The problem with the Axiom of Choice is that it lets the Four Horsemen loose in the land: it allows an initiate, for example (by an ingenious train of reasoning), to cut a golf ball into a finite number of pieces and put them together again to make a globe as big as the sun. Not only are its results an affront to intuition, but by not requiring us to know how we do the choosing, it adds to the Formalist shift of mathematics sideways.
