Why do we define vector spaces over a general $\mathbb{F}$ (field) rather than $\mathbb{C}$ (complex numbers)? In my linear algebra class, many times we define vector spaces over the field $\mathbb{F}$ (i.e. $\mathbb{F}^n$) and then prove things about them. The instructor has defined $\mathbb{F}$ as "either $\mathbb{C}$ or $\mathbb{R}$". Because $\mathbb{R}$ is a subset of $\mathbb{C}$, why can we not simply define vector spaces over $\mathbb{C}$ instead of introducing new notation? 
Isn't it true that a property holding over  $\mathbb{C}$ immediately tell us it holds over $\mathbb{R}$ because $\mathbb{R}\subset \mathbb{C}$?

There are different fields: Field of rational numbers, algebraic function fields, algebraic number fields, p-adic fields... 
The field is the set of scalars for the vector space. 
Is the division in $\mathbb{R}$ similar to that in $\mathbb{C}$? Absolute value? 
