This question is inspired by the other question which asked for a proof that $i^i$ is a real number.
Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he replied that it is impossible and that one only can postulate this by a convention.
So I wonder whether he is right? Is $0^0=1$ axiom really independent of all other axioms defining standard complex numbers and exponentiation?
UPDATE
It seems that no response so far tried to provide an answer to my clear question, that is whether $0^0=1$ is an independent axiom or not. Most answers are trying to defend particular values for $0^0$ which the authors prefer with some indirect or abstract argumentation, mostly involving taking limits or citing practical purpose.