Complex numbers as vector spaces

Question

I know that complex numbers can form a vector space over the field of real numbers as they obey the core axioms that constitute a vector space, however in vector spaces, the only operations that are defined are vector addition and scalar multiplication, which means vector multiplication isn’t defined, if that’s the case, then how come the multiplication of two complex numbers ( two vectors) can yield another complex number (i.e vector) ?

Am I missing something obvious ?

Note that I am well aware of the dot product and the cross product, so I’m talking about the normal vector multiplication that is not defined in a vector space

Thanks in advance

Answer 1

Complex numbers are a vector space over $\mathbb{R}$ and thus can be handled as vectors and used to solve geometric problems but they have a different "more extended" structure with multiplication, conjugate, and so on.

Answer 2

You're correct that not all vector spaces have vector multiplication. I don't see why you'd have a problem with a vector space having vector multiplication though. Nothing says that cannot happen.