Consider for example the function \begin{align*} f: (-2 \pi, 2 \pi) \times (-2 \pi, 2 \pi) \to \mathbb{C}, (x,y) \mapsto x^2 e^{i y} \end{align*} Note that this is not actually the function I am working with, but for the sake of simplicity let's consider the $f$ above. I now want to plot the image of this function, where both variables $x$ and $y$ vary over their entire domain. So the result should be the complex ball of radius $4 \pi^2$ centered at $0$.
If I had only one variable, I could just use complexplot, e.g.
complexplot(f(x,0), x=-6..6)
So what I could technically try is to fix certain values of $y$, plot the image for these values and then overlay all the (hundreds of) plots into one single plot. But this does not seem like the way to do it. Is there a more elegant way to plot the image of a complex-valued function depending on two real variables?
Most commonly what I see in books like An Atlas of Functions (K. Oldham, J. Myland, & J. Spanier, Springer, 2E, 2009) is a presentation of the real and imaginary parts in a 3-D plot. You might also think of plotting the magnitude and argument. I've shown the four examples in the figure below. You could also consider contour plots if you don't want go 3-D.