[mafuy]

Yes. Prominent example is the axiom of choice. There was a LOT of discussion about it by mathematicians. https://plato.stanford.edu/entries/axiom-choice/

For instance, with it, you can double the volume of a body by looking at it from a weird angle.

[SetTheorist]

That's a very misleading description of Banach-Tarski. You need to break up the body into a few (very weird) pieces and maneuver them (via only rigid motions - rotations and translations).