Real vector space. A real scalar $a \in \mathbb{R}$ acting on a 3-dimensional vector $\vec{v} \in \mathbb{R}^3$ can be visualized as the shrinking or stretching of $\vec{v}$ by a factor of $a$.
Complex vector space. Suppose now $z \in \mathbb{C}$. Is there a common or intuitive way to visualize what $z$ does to $\vec{v}$ in the case of
$$ z \vec{v} \in \mathbb{R}^3 $$
?
This nice notebook shows you how to map images using $f(z)=z^2$