Although you asked this question a while ago, I have a new answer.
You may be interested by the website reflex4you.com in "expert" mode. For example, the $cosh$ function is the following:
$z\rightarrow cosh(z)$ (click on the picture to visit the website and enter other formulas)
Although the axis are not shown, the window is between $-4-3i$ and $4+3i$. The zero is black and white is infinite. 1 is red and -1 is cyan. This representation is quasi-invariant if you take the negative of the picture, it will represent the inverse function.
To analyze this picture, you can consider that the values on the horizontal axis (the real axis) are red and darkest a zero, which means that $cosh(x) > cosh(0)$. The black spot are zeroes of the function corresponding to $+/- \pi/2$; Colored points denote complex functions. $i$ is greenish whereas $-i$ is purplish.
By clicking on the image above, you can modify the formula and try out other ones. I feel it is a very good way to represent complex functions.
To answer your original question, I wrote an algorithm usable on the website to compute the f-plane, for example for x constant. You can modify the value of the parameter c (constant for $x$). @M.Strochyk was right, these are ellipses.
let c = 1;
set k = 0;
let min = -4;
let max = 4;
let n = 100;
func f = cosh(z);
set result = 0;
let threshold = 0.1;
repeat n in
set theta = ((max-min)*k/n+min)*i+c;
set result = if(abs(f(theta)-z) < threshold, theta, result);
set k = k + 1;
Click on the picture to be able to modify it by entering different values.
DISCLAIMER: I am one of the co-authors of this website.