[lopsidedBrain]

It is partly historical notoriety. It allows one to prove certain statements that seem counter-intuitive. So proofs that employ it can often be viewed with some skepticism.

Moreover, this axiom is _independent_ of the other axioms in ZFC. It is in fact possible to have entirely self-consistent "worlds" of mathematics, ones where axiom of choice is true, and ones where it is false.

More details and examples of alternate axioms are in the Wikipedia article: https://en.wikipedia.org/wiki/Axiom_of_choice

If it seems weird that math can give you contradictory results, remember that the difference only shows up when you deal with some form of infinity (e.g. when performing an operation on an infinitely large set). For any usage of math in the real world, the truth or falsity of this axiom won't give you contradictory results.