In the following wikipedia article we read that :
Let $S$ be a vector space or an affine space over the real numbers, or, more generally, over some ordered field. This includes Euclidean spaces, which are affine spaces. A subset $C$ of $S$ is convex if, for all $x$ and $y$ in $C$, the line segment connecting $x$ and $y$ is included in $C$.
My question is why we need the field to be ordered in this definition? and does this mean that there is no notion of convex subsets in complex vector spaces?
The line segment connecting $x$ to $y$ is the set$$\{(1-\lambda)x+\lambda y\mid0\leqslant\lambda\leqslant1\}.$$As you can see from this definition, you are supposed to have an order relation on your field of scalars.
If $V$ is a complex vector space and $C\subset V$, we say that $C$ is convex if it is convex when we see $V$ as a real vector space.