f[x_,y_]=1/(1+(x+I y)*(x+I y))

Plot3D[Norm[f[x,y]],{x,-2, 2},{y,-2,2}]
Plot3D[Arg[f[x,y]],{x,-2, 2},{y,-2,2}]
Plot3D[Re[f[x,y]],{x,-2, 2},{y,-2,2}]
Plot3D[Im[f[x,y]],{x,-2, 2},{y,-2,2}]

f[x_,y_]=(x+I y)*(x+I y)

Plot3D[Norm[f[x,y]],{x,-2,2},{y,-2,2}]

f[x_,y_]=1/(1-Norm[(x+I y)]^2)

Plot3D[f[x,y],{x,-2,2},{y,-2,2}]

f[x_]=1/(1-x^2)

Plot[f[x],{x,-2,2}]

ParametricPlot3D[{r*Cos[u],r* Sin[u],f[r]},{u,0,2Pi},{r,0,2},Mesh->15,BoundaryStyle->Black,PlotStyle->FaceForm[Red,Yellow]]

ParametricPlot3D[{r*Cos[u],r* Sin[u],f[r]},{u,0,2Pi},{r,0,4},Mesh->15,BoundaryStyle->Black,PlotStyle->FaceForm[Red,Yellow]]
Trouble!  More trouble below....

Assuming[x ∈ Reals, Re[x]]

Assuming[x ∈ Reals, Im[x]]