Intuition for multiplying vectors by complex scalars There is a very pleasing interpretation of multiplying a vector by a real scalar. I can ''stretch or shrink'' and also ''flip over the origin (in the case of negative scalars)''.
Is there some analogous way to think of scalar multiplication when the vector field is over $\mathbb{C}$ instead? Intuitively I feel like its the same idea, just with more freedom over where scalars can point the vector (in the same way that complex number multiplication rotates and stretches other complex numbers) but I can't seem to write down anything concrete.
 A: The problem you're having is that a function with a complex input and a complex output requires four dimensions to graph.  The best we can do is consider the complex multiplier in the form $re^{i\theta}$ so that $r$ is the scalar multiplier you are used to, and the $e^{i\theta}$ part we have to imagine as a rotation through some angle $\theta$ in some invisible dimension.
This is often considered as a "phase" at every point, a function.
A: In the real case scalar multiplication takes a vector $v$ to another vector $w$ which lies in the linear subspace spanned by the vector $v$, which is a line, thus the multiplication stretches or shrinks the vector but does not change the line it spans. Similarly in the complex case, scalar multiplication takes a vector $v$ to another vector $w$ which lies in the complex span of $v$, this is now a 2-dimensional real vector space, therefore your intuition is correct, multiplying by a complex scalar can stretch, shrink or rotate the vector (where rotations are only possible in the plane spanned by the inital vector). You can think of it this way: given any vector $v\in\mathbb{C}^n$ and a scalar $a+ib=\lambda\in\mathbb{C}$ then multiplication by $\lambda$ changes $v$ to the vector $v = av+ibv$. Thus we can see that we can stretch or shrink $v$, but also add a shrunken or stretched version of $iv$ which gives the rotation.
A: Assume that you have
$$w=\lambda\,z\ ,\tag{1}$$
where everything is complex. In addition we imagine that $\lambda\in{\mathbb C}$ is constant, preferably $\ne0$, whereas $z$ is our independent variable, and $w$ changes when $z$ changes. Of course you can compute all of this algebraically through $\lambda=\mu+i\nu$,$\>z=x+iy$, $\>w=u+iv$.
But there is also a geometric interpretation of the transformation $z\mapsto w:=\lambda\,z$. Of course you have heard of such things when you first met complex numbers. You can draw a chosen point $z$ and its image point $w$ in the same Argand plane. The scaling factor $\lambda$ can be written in the form $\lambda=|\lambda|\,e^{i\phi}$ for a certain $\phi\in[0,2\pi[\>$. The equation $(1)$ then means that you obtain $w$ (resp., the vector $\vec{0w}$) from $z$ by scaling the vector $\vec{0z}$ by the $\geq0$ factor $|\lambda|$ and rotating this intermediate result by $\phi$ counterclockwise around $0$.
Of course I have just given another version of the rule "under complex multiplication the absolute values multiply, and the arguments add".
