Mesh Functions

Manipulate[ComplexPlot[z^3-z^2+1/z,{z,-1.4-1.4 I,1.4+1.4 I},

PlotPoints->50,Mesh->Automatic, MeshStyle->{Gray,White},
ImageSize->500,MeshFunctions->meshfunction],

{meshfunction,
{{Log10[Abs[#2]]&,Arg[#2]&},{Abs[Re[#2]]^.5&,Abs[Im[#2]]^.5&},
{Cos[#2]^.01&,Sin[#2]^.01&},
{Cos[Log10[Abs[#2]]]&,Sin[Log10[Abs[#2]]]&}}}]