Yup, I did a few projects on equivalents of AC back in the day. That's just my favorite "appeal to intuition" one (my favorite "appeal to intuition" against AC is: the identity function is the sum of two periodic functions (though this is a consequence and not equivalent)).

Equivalence breakdown in alternate systems is a wonderful topic. I've been trying for a couple years now to figure out how to get back into set theory now that I'm out of academia. Maybe later this summer...