ComplexPlot introduced in 12.0 can be used to create nice distorted checkerboards:
ComplexPlot[z (3 + Exp[Im[z]]), {z, 3}, Mesh -> 20, MeshFunctions -> {Re[#2] &, Im[#2] &}, MeshShading -> {{None, Black}, {Black, None}}, Frame -> False]
Here's an analytic function:
ComplexPlot[z Log[(1 + z^(11/2))/(1 - z^(11/2))], {z, 1.5}, Exclusions -> None, Mesh -> 22, MeshFunctions -> {Re[#2] &, Im[#2] &}, PlotPoints -> 100, MeshShading -> {{Automatic, None}, {None, Black}}, Frame -> False]
