Basically, if you live and work in the same state, you pay income tax to that state. It doesn't matter where the company that employs you is located.
I assume that you will be an employee (you will get a W-2), not a contractor. Based on the situation you described, the employer should not withhold North Carolina tax. You will work in Arizona, not in North Carolina. You will not have to pay any North Carolina tax or file a North Carolina tax return. All your income is for work that you do in Arizona. If possible, the company should withhold Arizona tax. If they can't do that, they should not withhold any state tax at all. In that case you will have to make quarterly estimated tax payments to Arizona yourself.
However, if you sometimes go to the company offices in North Carolina, your pay for the time that you are there is North Carolina income. You would then have to file a North Carolina nonresident tax return and pay North Carolina tax on that portion of your income. Since you live in Arizona, all of your income is subject to Arizona tax, no matter where the income is from. But you would get a credit on your Arizona resident tax return for part or all of the tax you pay to North Carolina, so you would not be double taxed.
There are a few state that have different rules for telecommuters or remote workers, but North Carolina is not one of them.
