Peter Humphries
2008 ASC report:
Several versions of "the" Banach-Tarski theorem, and
— possible sets of axioms — e.g. non-measurability & Hahn-Banach Theorem
James Taylor
— When mathematicians unify a theory, they discard definitions, and that has a lot of consequences, even though they tend to not say anything about those consequences.
— Problems which get blamed on AC could alternatively be blamed on
— definitions (e.g. Lesbegue measure), or
— ZF (see above e.g. James Taylor)
- example: Importance of topological invariant definition in proving the Poincar' Conjecture

Axiom of Choice could be a case study for this general point:
When e.g. a topologist wants to prove some theorem in topology, they'll be able to prove it in any reasonable set theory.
BUT in some set theories they'll need auxiliary axioms that they might not like (e.g. AC) and in other set theories they might not.