Planar complex analytic mapping

I would like to visualize planar complex analytic mappings

Points (x,y) will be mapped to points (u,v).
For example u=xx-yy and v=2xy would be the complex mapping f(z)=z^2.

Here is what I would like to achieve (don’t mind the theory text in Finnish):

I think I should find/make a suitable shader?
(Disclaimer. I am not a very experienced coder. But I have used JavaScript for education for 4 years and Three.js for 1 year.)

Are you also looking for solutions without custom shaders? If it is just this chessboard pattern to be transformed, I’d expect that JS can handle its modification in real time.

PS. The image in the link is approximation. The red squares should be mapped to curved trapezoids.