To me, the Axiom of Choice is absolutely natural – if we can pick one element of a set then we surely can pick elements from a collection of sets, who cares that this collection can be infinite.
The problem with this axiom is that it can't be formally proven from ZF axioms. But there are a lot of other natural things that can't be proven in ZF. For example, the consistency of ZF can't be proven in ZF due to Godel's incompleteness.
The problem with this axiom is that it can't be formally proven from ZF axioms. But there are a lot of other natural things that can't be proven in ZF. For example, the consistency of ZF can't be proven in ZF due to Godel's incompleteness.