New Mexico does not requires annual state or tax filing if you don't do business in the state.
When you open your LLC, your registered agent has to record your information in their book but they do not record it with the state. When I opened mine, I was never asked to provide any ID, I just needed to write a name and an address.
The only fee that you have to pay annually for a NM LLC is your registered agent (around $35 for the cheapest).
However, NM LLC does not have great asset protection statutes as the charging order is not the only︀ remedy. So you may want to have it owned by a WY holding LLC that︁ has excellent asset protection. But if you only are looking for anonymity, NM is a︂ great state to have your company registered in.