Don't take these axioms too seriously!  Math is not about axioms, despite what some people say.  Axioms are one way to think precisely, but they are not the only way, and they are certainly not always the best way.  Also, there are a number of ways to phrase these axioms, and different books will do this differently, but they are all equivalent (unless the book author was really sloppy).

Axioms of real vector spaces

A real vector space is a set X with a special element 0, and three operations:

These operations must satisfy the following axioms:
Axioms of a normed real vector space

A normed real vector space is a real vector space X with an additional operation:

This norm must satisfy the following axioms, for any x,y in X and any real number c:

Complex vector spaces and normed complex vector spaces are defined exactly as above, just replace every occurrence of "real" with "complex".  Note, though, that even in a complex vector space, the norm ||x|| is still a non-negative real number.
Thanks to Maxwell Davenport for a correction.