A mapping of the plane defines simply another point (u,v) in the plane as a function of the coordinates (x,y) of a point in the plane. The mapping is defined by the functions for the new coordinates u=f(x,y) and v=g(x,y). We can use any mixed polynomials of x and y. They need not be related to a complex function z(x+i*y).
Typically, the iterated points go to an attracting point. Then the starting point and the end point of the iteration do not have relevant information for coloring. Instead we use the sum of the angles φ=atan2(y,x) of the points at each iteration step. This is roughly the same as to the sum of the complex angles I used to compensate trivial phase changes in my earlier posts. Then, the color of a pixel at coordinates (x,y) results from this sum.
Alternatively, the iterated points go to infinity. Then we have to avoid overflow. Thus we stop the iterations if the distance of the point from the origin exceeds some limit. Then, we can get the color from the sum of the angles as above or we simply take the angle of the last iterated point.
It is fun to play with the functions and other parameters. You can get more or less symmetrical results:
Asymmetric results too can be interesting: