set theory - Decidability of the Axiom of Choice

There are, of course, exactly two ways for a theory to decide AC. Either it proves that AC is true, or it proves that AC is false.

Consequently, if you have a theory extending ZF and deciding AC, then either your theory includes ZFC or it includes ZF+¬AC. Thus, there are two minimal possibilities which meet your requirement, and any theory extending ZF and deciding AC must extend one of them.

The results of Godel on the constructible universe show that if ZF is consistent, then so is ZF+AC. And the results of Cohen on the forcing method show that if ZF is consistent, then so is ZF+¬AC. So both of these minimal theories are consistent, if ZF itself is consistent.

There are, of course, a huge variety of further extensions of ZFC that are intensely studied in set theory, and you can learn about them in any introductory graduate level set theory text. For example, one will want to know whether V=L, or whether CH holds, or GCH, or Diamond, whether there are Suslin trees or not, or large cardinals, and so on. Similarly, there are also a large variety of further extensions of ZF+¬AC that are studied, some quite intensely. For example, one might want to have the countable AC, or DC, or the Axiom of Determinacy and so on.