Conformal maps have the interesting property that they map circles and points to points and circle.
I wonder if the only holomorphic functions with this property are conformal maps. But I realized I don’t know any way to write “mapping lines and circles to lines and circles” as a functional equation which can then be solved or approximated for a collection of functions (which would in principle be the conformal maps).
The answer is that the functions that take lines and circles to lines and circles are precisely the Möbius transformations
$$f(z) = \dfrac{az+b}{cz+d},\quad \text{with} \ \ ad-bc\neq0.$$
I began writing an outline of the proof, but then I found this pdf, which is reasonably self-contained and goes into far more detail than I was willing to type here:
http://www.maths.qmul.ac.uk/~sb/CA_sectionIVnotes.pdf