Investor Rights Agreement Dated as of October 13, 2005

Exhibit 4.1

Investor Rights Agreement

Dated as of October 13, 2005

among

Energy XXI Acquisition Corporation (Bermuda) Limited

Sunrise Securities Corp.

and

Collins Stewart Limited

INVESTOR RIGHTS AGREEMENT

This Investor Rights Agreement (this “Agreement”) is made and entered into as of 13 October 2005 by and among Energy XXI Acquisition Corporation (Bermuda) Limited, a Bermudan company (the “Company”), and Sunrise Securities Corp. and Collins Stewart Limited (collectively, the “Placement Agents”), for the benefit of the holders (the “Holders”) from time to time of the Company’s common shares, par value $.001 per share (the “Common Shares”) and warrants (the “Warrants”), each Warrant for the purchase of one (1) Common Share.

Reference is made to the Company’s Offering Circular, dated 13 October 2005 (the “Offering Circular”), relating to the offer and sale of fifty million (50,000,000) units (the “Units”), each Unit consisting of one (1) Common Share and two (2) Warrants. For the benefit of the Holders and in consideration of the Placement Agents entering into a placing agreement with the Company for the placing of the Units, the Company has agreed to provide the investor rights set forth in this Agreement.

The parties hereby agree as follows:

SECTION 1. DEFINITIONS

As used in this Agreement, the following capitalized terms shall have the following meanings:

ADR: As defined in Section 2(b) hereof.

Commission: The U.S. Securities and Exchange Commission.

Exchange Act: The U.S. Securities Exchange Act of 1934, as amended.

NASD: National Association of Securities Dealers, Inc.

Offering: The offering of the Units pursuant to the Offering Circular.

Person: An individual, partnership, corporation, limited liability company, unincorporated organization, association, joint-stock company, trust, joint venture, government or any agency or political subdivision thereof or any other entity.

Placing Agreement: The placing agreement between the Company, Sunrise Securities Corp., Collins Stewart and the Company’s directors dated on or about the date of this Agreement.

Qualified Business Combination: A business combination which, when combined with the Company’s previous business combinations, results in the release of at least fifty percent (50%) of the initial amount of the trust fund (as more fully described in the Offering Circular).

Securities Act: The U.S. Securities Act of 1933, as amended.

‘34 Act Liquidated Damages: As defined in Section 5(a) hereof.

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‘34 Act Registration Deadline: The date of consummation of a Qualified Business Combination or, alternatively, in the case where no Qualified Business Combination shall have occurred but a business combination shall have been completed and the balance of the funds in the trust fund have been distributed to the new shareholders, the date of such distribution.

‘34 Act Registration Default: As defined in Section 4(a) hereof.

‘34 Act Registration Statement: A registration statement of the Company on Form 10 (or such other form which it is appropriate to use to register the Common Shares or ADRs under the Exchange Act), including all amendments and supplements thereto and all exhibits and material incorporated by reference therein.

SECTION 2. EXCHANGE ACT REGISTRATION AND LISTING

(a) The Company shall (i) no later than the date that is one hundred eighty (180) days following the ‘34 Act Registration Deadline cause to be filed with the Commission a ‘34 Act Registration Statement, and (ii) use its commercially reasonable efforts to cause such ‘34 Act Registration Statement to be declared effective on or prior to the date that is two hundred seventy (270) days following the ‘34 Act Registration Deadline.

(b) As promptly as practicable after the ‘34 Act Registration Statement shall have been declared effective, the Company shall use its commercially reasonable efforts to cause the Common Shares or American Depositary Receipts representing Common Shares (“ADRs”) to be authorized to be quoted and/or listed (to the extent applicable) on the American Stock Exchange, the New York Stock Exchange, the National Association of Securities Dealers, Inc. Automated Quotation System or the Nasdaq National Market (or, in each case, a successor thereto) or a similarly recognized national trading platform, if the Common Shares or ADRs so qualify.

SECTION 3. WARRANT SHARE REGISTRATION

(a) The Company shall file with the Commission, no later than the 60th day after the ‘34 Act Registration Statement is declared effective, a shelf registration statement (the “Shelf Registration Statement”) pursuant to Rule 415 under the Securities Act of 1933, as amended (the “Securities Act”) covering the Common Shares as and when issued upon exercise of the Warrants, including the Warrants underlying that certain unit purchase option dated October 13, 2005 (“Purchase Option”) issued by the Company to Sunrise Securities Corp. and/or its designees (the “Warrant Shares”). The Company shall (i) use commercially reasonable efforts to cause the Shelf Registration Statement to be declared effective under the Securities Act no later than 105 days after the ‘34 Act Registration Statement is declared effective unless the Commission reviews such registration statement in which case the 105 day period shall be increased to 120 days, and (ii) use commercially reasonable efforts to keep the Shelf Registration Statement effective until the earlier of (x) the date when the holders of the Warrants and the Warrant Shares are able to sell such shares immediately without restriction pursuant to volume limitation provisions of Rule 144 under the Securities Act; and (y) such time as all of the Warrant Shares have been sold either pursuant to the Shelf Registration Statement or pursuant to Rule 144 under the Securities Act or any similar provision then in force.

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(b) The Company shall use commercially reasonable efforts to cause the prospectus contained within the Shelf Registration Statement to be usable for its intended purpose during the Designated Period (as defined below) including filing any amendments and supplements to the Shelf Registration Statement or the prospectus as may be required. The “Designated Period” shall mean the period beginning two days after the public announcement of quarterly and year-end earnings by the Company and ending 30 days thereafter, provided that the relevant earnings announcement shall in no event be issued publicly after November 29, March 1, May 30 and October 28, in the case of the first quarter, second quarter, third quarter and fiscal year-end, respectively.

(c) The Company may suspend the effectiveness of the Shelf Registration Statement or the use of the prospectus that is part of the Shelf Registration Statement during specified periods during the Designated Period under certain circumstances relating to pending corporate developments, public filings with the Commission and similar events; provided, that due to concerns about possibly disseminating material non-public information, the Company need not specify the nature of the event giving rise to a suspension in any notice to holders of the Warrant Shares of the existence of such a suspension. Any such suspension period shall not exceed an aggregate of 45 days of the Designated Periods in any 360-day period. The number of days during which the Company may have suspended use of the prospectus during the Designated Periods will be added to the end of the Designated Period with respect to the next one or more successive quarterly periods or, when the prospectus has been appropriately amended or supplemented to reflect the event or events giving rise to a suspension, the amended or supplemented prospectus shall be delivered to the selling Holders named in the Shelf Registration Statement and they shall have a period of time equal to the number of days during which selling was suspended to sell Warrant Shares using the amended or supplemented prospectus.

(d) In connection with the preparation of the Shelf Registration Statement, the Company shall mail a questionnaire to all warrantholders and holders of restricted Warrant Shares of record as of the close of business on the day the ‘34 Act Registration Statement becomes effective. In order to be named as a selling shareholder in the prospectus at the time of effectiveness of the Shelf Registration Statement, such Holder must complete and deliver the questionnaire to the Company on or prior to the 15th business day after the questionnaire is mailed. Holders not named as selling shareholders therein will not be able to avail themselves of such Shelf Registration Statement. Accordingly, transferees of warrants or Warrant Shares may not be able to avail themselves of registration; provided, that, every six months after the Shelf Registration Statement becomes effective, the Company shall mail to all warrantholders of record and holders of restricted Warrant Shares of record a new questionnaire and amend or supplement the Shelf Registration Statement to include those that have duly completed and returned a questionnaire to the Company. The Company may, in its sole discretion, permit more frequent supplements to the Shelf Registration Statement to include transferees. Failure to timely provide a duly completed questionnaire will result in the non-responding security holders exclusion from the Shelf Registration Statement or the use of the prospectus that is part of the Shelf Registration Statement. Any Holder who has properly elected to be included in the Shelf Registration Statement will (i) be required to be named as a selling stockholder in the related prospectus, (ii) be required to deliver a prospectus to purchasers, and (iii) be subject to the civil liability provisions under the Securities Act in connection with any sales pursuant to such Shelf Registration Statement.

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SECTION 4. “PIGGYBACK” REGISTRATIONS

(a) If at any time the Company shall propose to register under the Securities Act (other than pursuant to Sections 2 and 3 of this Agreement) any of its Common Shares for the account of its Existing Shareholders (as defined in the Offering Circular), it will promptly give written notice to all Holders of Common Shares that are (i) holders of Common Shares bearing a restrictive legend that are not freely saleable in the United States under Rule 144(k) of the Securities Act, and (ii) that are known by the Company to be affiliates of the Company (other than officers and directors of the Company) (such Common Shares hereinafter referred to as “Registrable Shares”) of its intention so to do. Upon the written request of any such holder, received by the Company within 20 days after the giving of any such notice by the Company, to register any or all of its Registrable Shares, the Company will use its reasonable best efforts to cause the Registrable Shares as to which registration shall have been so requested to be included in the securities to be covered by the registration statement proposed to be filed by the Company, all to the extent requisite to permit the sale or other disposition by the holder (in accordance with its written request) of such Registrable Shares so registered.

(b) If the registration of which the Company gives notice as provided above is for a registered public offering involving an underwriting, the Company shall so advise the Holders of Registrable Shares as a part of the written notice given pursuant to this Section 4. In such event, the right of any Holder of Registrable Shares to registration pursuant to this Section 4 shall be conditioned upon such Holder’s participation in such underwriting to the extent provided herein. All Holders of Registrable Shares proposing to distribute their securities through such underwriting shall (together with the Common Shares to be registered by the Company in such registration (the “Other Shareholders”)) enter into an underwriting agreement in customary form with the underwriter or underwriters selected for underwriting by the Company. If any Holder of Registrable Shares disapproves of the terms of any such underwriting, it may elect to withdraw therefrom by written notice to the Company and the underwriter. Any Registrable Shares or other securities excluded or withdrawn from such underwriting shall be withdrawn from such registration.

(c) Notwithstanding any other provision of this Section 4, if the underwriter determines that marketing factors require a limitation on the number of shares to be underwritten, the underwriter may exclude from such registration and underwriting all or a portion of the Registrable Shares which would otherwise be underwritten pursuant to this Section 4. The Company shall so advise all Holders of securities requesting registration of any limitations on the number of shares to be underwritten and the number of shares of securities that are entitled to be included in the registration; provided, that the number of shares to be underwritten shall be allocated pro rata among all Holders of Registrable Shares and the Existing Shareholders in proportion, as nearly as practicable, to the respective amounts of Registrable Shares owned by them.

SECTION 5. LIQUIDATED DAMAGES

(a) If the ‘34 Act Registration Statement shall not have been declared effective on or prior to the date that is two hundred seventy (270) days following the ‘34 Act Registration Deadline (a “‘34 Act Registration Default”), the Company shall, as promptly as practicable and in no event later than two (2) days following the end of the month in which such ‘34 Act Registration Default occurred, pay to each Holder of Common Shares liquidated damages (“‘34 Act Liquidated Damages”) in the form of Common Shares in an amount equal to 0.5% of the number of such Holder’s Common Shares held on the date of such ‘34 Act Registration Default; provided, that a Holder shall be paid ‘34 Act Liquidated Damages only with respect to (i) any Common Shares that were acquired by such Holder upon the

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consummation of the Offering or subsequent to the Offering (if such Common Shares were originally offered as part of the Offering) and (ii) any Common Shares that were acquired by such Holder pursuant to the exercise of the Purchase Option. The Company shall pay additional ‘34 Act Liquidated Damages within two days of the end of each month until the ‘34 Act Registration Default shall have been cured; provided that a pro rata portion of the ‘34 Act Liquidated Damages shall be paid with respect to any month in which the Company shall have been in ‘34 Act Registration Default for a portion of such month; and provided, further, that ‘34 Act Liquidated Damages shall be payable for a maximum period of twelve (12) months following the occurrence of the ‘34 Act Registration Default.

(b) Each of the following is hereafter referred to as a “Warrant Registration Default”: (i) the Shelf Registration Statement has not been filed with the Commission within 60 days after the ‘34 Act Registration Statement is declared effective; or (ii) the Shelf Registration Statement has not been declared effective prior to or on the 105th day or the 120^th day, as applicable, following the day the ‘34 Act Registration Statement is declared effective (the “Effectiveness Target Date”); or (iii) at any time after the Shelf Registration Statement becomes effective the Shelf Registration Statement ceases to be effective or fails to be usable and (x) the Company does not cure such failure by the start of the next Designated Period by a post-effective amendment, prospectus supplement or report filed pursuant to the Exchange Act, and (y) the period during the Designated Period of the suspension period, when aggregated with other suspension periods during Designated Periods for the prior 360-day period, aggregates more than 45 days.

(c) If a Warrant Registration Default occurs, liquidated damages will accrue from and including the day following the Warrant Registration Default to but excluding the earlier of (i) the day on which the Warrant Registration Default has been cured, and (ii) the date the Shelf Registration Statement is no longer required to be kept effective. Liquidated damages pursuant to a Warrant Registration Default shall be (w) paid to each Holder of Warrants in the form of Common Shares in an amount equal to 0.5% of the number of Common Shares underlying each Warrant Holder’s Warrants; (x) payable promptly after the occurrence of the Warrant Registration Default, but in no event later than two days after the end of the month in which the Warrant Registration Default occurred; (y) payable within two days of the end of each month, until the Warrant Registration Default has been cured, provided that a pro rata payment shall be made with respect to a month, a portion of which, the Company has been in default; and (z) payable for a maximum of the lesser of (1) 12 months, or (2) a number of months equal to 12 minus the number of months for which the Company has paid ‘34 Act Liquidated Damages.

Notwithstanding any other provision of this Agreement, the liquidated damages contemplated in this Section 5 shall be the sole and exclusive remedy available to the Holders in the event of any failure by the Company to comply with the terms of this Agreement.

SECTION 6. CURRENT PUBLIC INFORMATION

(a) From and after the date hereof and until the ‘34 Act Registration Statement shall have become effective, the Company shall furnish to the Holders:

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provided, however that save to the extent required by law to be included in the Commission reports referred to above, the reports set forth above shall not be required to (i) contain any certification required by the Sarbanes-Oxley Act, (ii) include any exhibits, (iii) contain quarterly reconciliations to GAAP, (iv) include separate financial statements for any acquired businesses or (v) any pro forma financial information.

(b) With respect to any proxy statement required to be prepared by the Company in connection with a business combination, such proxy statement shall contain substantially the same information as would be required in a proxy statement for an acquisition filed with the Commission provided, however, that the financial statements included in such proxy statement, if permitted by applicable AIM requirements, may be unaudited and may be prepared in accordance with IFRS and reconciled to GAAP. In addition, such proxy statement shall describe the implications of approving a business combination, including whether the Warrants become exercisable upon consummation of the business combination and the following operational information with respect to the acquisition target: i) the production profile and historical oil and gas production decline figures; ii) the results of a third party reserve report; iii) management’s estimates of effective lifting costs per barrel oil equivalent; and iv) the previous owner’s three-year effective lifting cost history plus an additional two years to the extent available.

SECTION 7. REGISTRATION EXPENSES

(a) All expenses incident to the Company’s performance of or compliance with this Agreement will be borne by the Company regardless of whether any registration statement hereunder becomes effective, including without limitation and as applicable: (i) all Commission, securities exchange or NASD registration and filing fees and expenses; (ii) all fees and expenses of compliance with U.S. federal securities and state blue sky laws and compliance with the rules of a securities exchange or NASD, as applicable; (iii) all expenses of printing, messenger and delivery services; (iv) all fees and disbursements of counsel for the Company; and (v) all fees and disbursements of independent certified public accountants of the Company (including the expenses of any special audit and comfort letters required by or incident to such performance) but excluding those fees borne by other parties as detailed in the Placing Agreement and any portion of commissions or discounts attributable to the Common Shares being offered and sold by the Holders.

(b) The Company will, in any event, bear its internal expenses (including, without limitation, all salaries and expenses of its officers and employees performing legal or accounting duties), the expenses of any annual audit and the fees and expenses of any Person, including special experts, retained by the Company.

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SECTION 8. INDEMNIFICATION

(a) The Company shall indemnify and hold harmless each Holder, its officers and employees and each Person, if any, who controls any such Holders, within the meaning of the Securities Act, from and against any loss, claim, damage or liability, joint or several, or any action in respect thereof, to which that Holder, officer, employee or controlling Person may become subject, under the Securities Act or otherwise, insofar as such loss, claim, damage, liability or action arises out of, or is based upon (i) any untrue statement or alleged untrue statement of a material fact contained in any registration statement filed pursuant hereto or in any amendment or supplement thereto; (ii) the omission or alleged omission to state in any registration statement filed pursuant hereto or in any amendment or supplement thereto any material fact required to be stated therein or necessary to make the statements therein, in light of the circumstances under which they were made, not misleading, and shall reimburse each Holder and each such officer, employee or controlling Person promptly upon demand for any legal or other expenses reasonably incurred by that Holder, officer, employee or controlling Person in connection with investigating or defending or preparing to defend against any such loss, claim, damage, liability or action as such expenses are incurred provided that no Holder shall be indemnified for misstatements or omissions arising from information provided by the Holder in writing for inclusion in any registration statement filed pursuant hereto. The foregoing indemnity agreement is in addition to any liability which the Company may otherwise have to any Holder or to any officer, employee or controlling Person of that Holder.

(b) Promptly after receipt by an indemnified party under this Section 8 of notice of any claim or the commencement of any action, the indemnified party shall, if a claim in respect thereof is to be made against the Company under this Section 8, notify the Company in writing of the claim or the commencement of that action; provided, however, that the failure to notify the Company shall not relieve it from any liability which it may have under this Section 8 except to the extent it has been materially prejudiced by such failure; and provided, further, that the failure to notify the Company shall not relieve it from any liability which it may have to an indemnified party otherwise than under this Section 8. If any such claim or action shall be brought against an indemnified party, and it shall notify the Company thereof, the Company shall be entitled to participate therein and, to the extent that it wishes to assume the defense thereof with counsel reasonably satisfactory to the indemnified party. After notice from the Company to the indemnified party of its election to assume the defense of such claim or action, the Company shall not be liable to the indemnified party under this Section 8 for any legal or other expenses subsequently incurred by the indemnified party in connection with the defense thereof other than reasonable costs of investigation; provided, however, any indemnified party shall have the right to employ separate counsel in any such action and to participate in the defense thereof but the fees and expenses of such counsel shall be at the expense of the indemnified party unless (i) the employment of such counsel has been specifically authorized by the Company in writing, or (ii) such indemnified party shall have been advised by such counsel that there may be one or more legal defenses available to it which are different from or additional to those available to the Company and in the reasonable judgment of such counsel it is advisable for such indemnified party to employ separate counsel or (iii) the Company has failed to assume the defense of such action and employ counsel reasonably satisfactory to the indemnified party, in which case, if such

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indemnified party notifies the Company in writing that it elects to employ separate counsel at the expense of the Company, the Company shall not, in connection with any one such action or separate but substantially similar or related actions in the same jurisdiction arising out of the same general allegations or circumstances, be liable for the reasonable fees and expenses of more than one separate firm of attorneys (in addition to one local counsel) at any time for all such indemnified parties, which firm shall be designated in writing by the Holders of a majority in principal amount of the outstanding Common Shares and the Common Shares issuable upon exercise of the Warrants. The Company shall not (i) without the prior written consent of the indemnified parties (which consent shall not be unreasonably withheld), settle or compromise or consent to the entry of any judgment with respect to any pending or threatened claim, action, suit or proceeding in respect of which indemnification or contribution may be sought hereunder (whether or not the indemnified parties are actual or potential parties to such claim or action) unless such settlement, compromise or consent includes an unconditional release of each indemnified party from all liability arising out of such claim, action, suit or proceeding or (ii) be liable for any settlement of any such action effected without its written consent (which consent shall not be unreasonably withheld), but if settled with the consent of the Company or if there be a final judgment of the plaintiff in any such action, the Company agrees to indemnify and hold harmless any indemnified party from and against any loss or liability by reason of such settlement or judgment.

(c) If the indemnification provided for in this Section 8 shall for any reason be unavailable to or insufficient to hold harmless an indemnified party under Section 6(a) in respect of any loss, claim, damage or liability, or any action in respect thereof, referred to therein, then the Company shall, in lieu of indemnifying such indemnified party, contribute to the amount paid or payable by such indemnified party as a result of such loss, claim, damage or liability, or action in respect thereof, (i) in such proportion as shall be appropriate to reflect the relative benefits received by the Company, on the one hand, and the Holders on the other, from the sale of the Common Shares or (ii) if the allocation provided by clause (i) above is not permitted by applicable law, in such proportion as is appropriate to reflect not only the relative benefits referred to in clause (i) above but any other relevant equitable considerations. The Company and the Holders agree that it would not be just and equitable if contributions pursuant to this Section 8(c) were to be determined by pro rata allocation (even if the Holders were treated as one entity for such purpose) or by any other method of allocation which does not take into account the equitable considerations referred to herein. The amount paid or payable by an indemnified party as a result of the loss, claim, damage or liability, or action in respect thereof, referred to above in this Section 8 shall be deemed to include, for purposes of this Section 8(c), any legal or other expenses reasonably incurred by such indemnified party in connection with investigating or defending any such action or claim. Notwithstanding the provisions of this Section 8(c), no Holder shall be required to contribute any amount in excess of the amount by which the net proceeds received by it in connection with its sale of Common Shares exceeds the amount of any damages which such Holder has otherwise paid or become liable to pay by reason of the untrue or alleged untrue statement or omission or alleged omission. No Person guilty of fraudulent misrepresentation (within the meaning of Section 1l(f) of the Securities Act) shall be entitled to contribution from any Person who was not guilty of such fraudulent misrepresentation. The Holders’ obligations to contribute as provided in this Section 8(c) are several and not joint.

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SECTION 9. MISCELLANEOUS

(a) No Inconsistent Agreements. The Company shall not, on or after the date of this Agreement, enter into any agreement with respect to its securities that is inconsistent with the rights granted to the Holders in this Agreement or otherwise conflicts with the provisions hereof. Except as disclosed in the Offering Circular, the Company has not previously entered into any agreement granting any registration rights with respect to its Common Shares to any Person except for the registration rights given to the initial investors and the placement agents. The rights granted to the Holders hereunder do not in any way conflict with and are not inconsistent with the rights granted to the holders of the Company’s Common Shares under any agreement in effect on the date hereof.

(b) Amendments and Waivers. The provisions of this Agreement may not be amended, modified or supplemented, and waivers or consents to or departures from the provisions hereof may not be given unless the Company has obtained the written consent of Holders of a majority of the principal amount of the outstanding Common Shares and the Common Shares issuable upon the exercise of the Warrants affected by such amendment, modification, supplement, waiver or consent.

(c) Notices. All notices and other communications provided for or permitted hereunder shall be made in writing by hand-delivery, first-class mail (registered or certified, return receipt requested), telex, facsimile or air courier guaranteeing overnight delivery:

(i) if to a Holder, at the address set forth in the record books of the Company; and

if to the Company to:

Energy XXI Acquisition Corporation (Bermuda) Limited

Canon’s Court

22 Victoria Street, PO Box 1179

Hamilton HM EX Bermuda

with a copy to:

Mr. John D. Schiller

c/o 1021 Main, Suite 2626

Houston, Texas 77002

with copies to:

Sunrise Securities Corp.

641 Lexington Avenue

New York, NY 10022

Attn: President

and

Collins Stewart Limited

9th Floor

88 Wood Street

London EC2V 7QR

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Any such notices and communications shall take effect at the time of receipt thereof.

(d) Successors and Assigns. This Agreement shall inure to the benefit of and be binding upon the successors and assigns of each of the parties.

(e) Counterparts. This Agreement may be executed in any number of counterparts and by the parties hereto in separate counterparts, each of which when so executed shall be deemed to be an original and all of which taken together shall constitute one and the same agreement.

(f) Headings. The headings in this Agreement are for convenience of reference only and shall not limit or otherwise affect the meaning hereof.

(g) Governing Law. THIS AGREEMENT SHALL BE GOVERNED BY, AND CONSTRUED AND INTERPRETED, IN ACCORDANCE WITH THE LAW OF THE STATE OF NEW YORK.

(h) Severability. In the event that any one or more of the provisions contained herein, or the application thereof in any circumstance, is held invalid, illegal or unenforceable, the validity, legality and enforceability of any such provision in every other respect and of the remaining provisions contained herein shall not be affected or impaired thereby.

(i) Entire Agreement. This Agreement is intended by the parties as a final expression of their agreement and intended to be a complete and exclusive statement of the agreement and understanding of the parties hereto in respect of the subject matter contained herein. There are no restrictions, promises, warranties or undertakings, other than those set forth or referred to herein with respect to the investor rights granted by the Company herein. This Agreement supersedes all prior agreements and understandings between the parties with respect to such subject matter.

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IN WITNESS WHEREOF, the parties have executed this Agreement as of the date first written above.