No, you cannot get a New Mexico driver's license if you currently have an Arizona driver's license that is suspended.
All 50 states within the United States (and the District of Columbia) look at the Problem Driver Pointer System (PDPS) and National Driver Register (NDR) when a person applies for a driver's license. The PDPS searches the National Driver Register (NDR), a central database of information maintained by the National Highway Traffic Safety Administration (NHTSA) on drivers who have had their licenses revoked or suspended or have been convicted of serious traffic violations such as driving while impaired by alcohol or drugs, and alerts the DMV doing the search of any issues a driver's license applicant may have in other states. Again, all 50 states and DC participate in placing information into the NDR and all will also search it to look for problem drivers.
If you plan to move to NM and apply for a driver's license there they will check the NDR or PDPS. The PDPS points the inquiring licensing bureau to the State of Record, where an individual's driver status and history information is stored. This helps with the basic tenet of each driver nationwide having only one driver license and one record. The federal government is trying to keep motorists from "shopping" around to get a driver's license when they have an active suspension or revocation.
The Motor Vehicles Division (MVD) of New Mexico states for new residents that once you have established residency in New Mexico, you are required to surrender your license from any other state and apply for a New Mexico driver's license. However a NM driver's license will not be issued if your driving privileges are suspended, revoked or denied by this state or any other state.
The NM MVD says you must first resolve the suspension or revocation of your license if you have one out of state before you can apply and obtain a driver's license from New Mexico. So New Mexico flatly states that you will need to take care of your suspension in Arizona before they will allow you to obtain a driver's license there. If you try to apply for a license without mentioning the AZ license issue the NM MVD will check the NDR and find the suspension and then deny your license any way.
When you get your license reinstated in Arizona you should then be able to apply and obtain a driver's license in New Mexico. In either state CarInsurance.com can help you check car insurance prices and find you the best car insurance for your situation.