Complex Domain coloring vs Complex Range (over the output) coloring: which one brings more information?

Question

Currently I am learning how to do domain coloring visualizations. Basically, as Wikipedia says:

Domain coloring is a technique for visualizing functions of a complex variable ...There were many earlier uses of color to visualize complex functions, typically mapping argument (phase) to hue.

So basically I have learned how to do the basic hue map when we have a complex injective function $f(z)$. These are some simple examples:

First row left-right: $(1)\ f(z)=\frac{1}{z}\ ,\ (2)\ f(z)=\frac{(z^2-1)(z-2-i)^2}{z^2+2+2i}\ ,\ (3)\ f(z)=z^3+1 $

Second row left-right: $(4)\ f(z)=z^5-1\ ,\ (5)\ f(z)=e^z$.

They are the mappings we are used to see. But to my surprise, then I tried a "reverse" mode, instead of mapping over the positions of the input complex points (the domain of the function), mapped the output points with the conversion to hue of the original input points, a kind of "Range coloring" or "Image coloring" or "coloring map over the result" of the function, the results were also very impressive. They are the other side of the coin of the relationship between the Domain and the Image/Range of the function:

Here is a zoom of the second example:

So my point is: I can understand that visualizing the Image/Range/output of the function in the position of the original points of the Domain as "hue/brightness/saturation" colors provides interesting information about the location of output of the function; but, why the Range or Image coloring is not as popular as the direct Domain coloring? or why are not they treated as complementary to each other?

So the questions I would like to share are:

1. Why the Range/Image coloring is not so popular as the Domain coloring technique?

2. What type of complementary information can we find in the Range/Image coloring that we could not find or understand by visualizing a Domain coloring pattern?

3. Are there papers or documents regarding this kind of alternative visualization?

I think that in terms of complex domain coloring still there are things to explode and we are seeing only one side of the coin yet. And the other side seems quite interesting!

(If somebody wants a copy of the Python code used to make the examples, please let me know and I will add it to the question)

Answer 1

Just in order to be able to close the question, I am gathering the kind comments made by Chris Culter and reuns. I think they give very valuable insights (written in the same time order they were written):

Credits to Chris Culter:

Not an answer, but an observation: If you think of an image as a map from a surface into a color space, then the "complex argument to RGB hue" mapping is a function $h:\Bbb C \to {[0,1]}^3$. Now given a function $f:\Bbb C_d \to \Bbb C_r$, the only way to compose $f$ and $h$ is to take $h∘f:C_d \to {[0,1]}^3$, which is the domain coloring method. To the extent that this is the preferred way to think about images, then domain coloring seems more natural.

Credits to reuns:

Take a function $\varphi: \Bbb C \to RGB={[0,1]}^3$. In one case you are plotting $(x,y) \to \varphi (f(x+iy))$. In the other case you are plotting $(x,y) \to \varphi(f−1(x+iy))$. Try with $f(z)=e^z$ to see why the fact $f^{−1}$ is not globally analytic means you'll have some problems to plot it in a satisfying way. Locally (if $f′(a)≠0$) then $f{−1}$ is just an analytic function, exactly as $f$.