I work for a company where employees are all full-time AZ residents. We have projects in NM in which we have sent them. Some are there for 2-3 weeks at a time living in corporate housing and some that go 4 days a week and live in a hotel/extended stay and return to AZ at end of week. One is now only going 2 days a week every other week or just as needed. What determines the state that the tax should be paid? And, when do you disconitnue the NM tax being paid and move it back to AZ? The one going only a couple of days a week is still having NM tax withheld. I don't believe any of them are being taxed by both NM and AZ. It looks like it is just NM tax. I have similar situation in New Jersey and Utah. If they were only taxed in one state and not in AZ do they file a non-resident tax form for this work in other states? They worked part of the year in AZ as well so their W-2 will show AZ taxes too can they file for a credit on the other state taxes?

During the periods you worked in NM, your income is NM income to be reported on NM tax return. File NM tax return if your NM income is now below the NM filing requirements.

On AZ tax return, you will report your worldwide income including NM income. Then you will claim credit for taxes paid to NM.
Your U.S. Tax Return: Working or Living in Two or More states (http://taxipay.blogspot.com/2008/06/working-in-two-or-more-states.html)