S Corp in a different state
You certainly could open an S-Corp in Wyoming, Nevada, or any of the United States, regardless of where your business address would be. (For example, you could use your New Mexico home address as the business address for your S-Corp registered in Wyoming).
One requirement to form a company in any state is maintaining a Registered Agent in that state for service of process (P.O. Boxes are not permitted), even though a business address may be just about anywhere internationally. We offer a registered agent service for those situations where a business owner does not reside in the same state as their business.
Keep in mind that even if you form your company in another state but your business would have physical presence in New Mexico you would still be required to register a Foreign Entity in New Mexico. To lean more about when it would be required to form a foreign entity you are welcome to read our answer to the question Doing business in another state.
Also, I wanted to point you to an interesting fact about your state - New Mexico LLCs are known to be very good asset protection tools, since the state does not require you to submit the names of members and there is no annual filing requirement (so once you form the company its there pretty much for good). You might consider NM LLC as an alternative for forming corporation in any other state (especially because you can elect the LLC to be taxed as S- or C-Corporation if thats what your accountant suggests).