Question

what's the craziest consequence of Zorn's lemma u know?

Answer 1

Not sure what you mean by "craziest" (most counterintuitive?), but I'll tell you a very important one that links algebra to geometry: in an arbitrary nonzero commutative ring, the intersection of all prime ideals is the set of nilpotent elements. So if an element of a nonzero commutative ring vanishes modulo all prime ideals, then that element might not be 0 but at least some power of it is 0.

Answer 2

The algebraic closure of the field of p-adic numbers Q_p is isomorphic to C.

This is so bizarre that many well-known mathematicians prefer to avoid it. For example, morphisms from the algebraic closure of Q_p to C appear in an essential way in Deligne’s second paper on the Weil conjectures and there he remarks that we prefers not to believe in such an isomorphism.

Answer 3

The Well-ordering Theorem.

Answer 4

The axiom of choice. Such a ridiculous property  
:)