N.M. Stat. § 62-11-6
The pendency of an appeal shall not of itself stay or suspend the operation of the order of the commission, but, during the pendency of such proceedings, the supreme court in its discretion may stay or suspend, in whole or in part, the operation of the commission's order on such terms as it deems just and in accordance with the practice of courts exercising equity jurisdiction. If the supreme court stays or suspends the operation of an order of the commission, pending a review of the order, the court may require the party seeking the stay or suspension to secure the other parties against loss due to the delay in the enforcement of the order, in case the order on appeal is affirmed, in such amount and in such form as the supreme court shall direct.
NMS § 62-11-6