The right to the compensation provided for in this act [N.M. Stat. Ann. Chapter 52, Article 1], in lieu of any other liability whatsoever, to any and all persons whomsoever, for any personal injury accidentally sustained or death resulting therefrom, shall obtain in all cases where the following conditions occur:

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A. at the time of the accident, the employer has complied with the provisions thereof regarding insurance;

B. at the time of the accident, the employee is performing service arising out of and in the course of his employment; and

C. the injury or death is proximately caused by accident arising out of and in the course of his employment and is not intentionally self-inflicted.