Geometrically, what is a complex vector space?
By definition I know it's a vector space which has scalar multiplication defined over the field C complex numbers. But what does this look like geometrically?
I know any point in $\mathbb{R}^2$ written as Cartesian coordinates $(x,y)$ can also be represented using imaginary numbers $x + iy$. But what is a complex vector space?
Aren't vector spaces already a form of representation of higher-dimensional spaces?
This is very open-ended; I guess I'm looking for some kind of intuition, a way to visualise this really. If anyone has relevant resources I'd welcome those too.
