f[z_] := ((z^2 - 1) (z - 2 - I)^2)/(z^2 + 2 + 2 I);
ContourPlot[Evaluate[#[f[x + I y]]], {x, -3, 3}, {y, -3, 3},
ColorFunction -> "DarkRainbow", Contours -> 20] & /@ {Re, Im, Abs,
Arg}
Plot3D[Evaluate[#[f[x + I y]]], {x, -3, 3}, {y, -3, 3},
ColorFunction -> "DarkRainbow", MeshStyle -> Opacity[.5],
MeshFunctions -> (#3 &), Mesh -> 20,
PlotStyle -> Opacity[.7]] & /@ {Re, Im, Abs, Arg}