Code:

f[z_, k_] := Log[Abs[z]] +
I*(Arg[z] + 2*k*Pi);
z1 = 2 - 2*I;
point1 = Graphics3D[{{PointSize[0.04],
Green, Point[{{Re[z1], Im[z1],
Im[f[z1, 0]]}}]}, {PointSize[0.04],
Blue, Point[{{Re[z1], Im[z1],
Im[f[z1, 1]]}}]}, {PointSize[0.04],
Red, Point[{{Re[z1], Im[z1],
Im[f[z1, -1]]}}]}}];
p1 = ParametricPlot3D[
{Re[z], Im[z], Im[f[z, 0]]} /.
z -> r*Exp[I*t], {r, 0, 3},
{t, -Pi, Pi}, PlotStyle -> Red]
p2 = ParametricPlot3D[
{Re[z], Im[z], Im[f[z, -1]]} /.
z -> r*Exp[I*t], {r, 0, 3},
{t, -Pi, Pi}, PlotStyle -> Blue]
p3 = ParametricPlot3D[
{Re[z], Im[z], Im[f[z, 1]]} /.
z -> r*Exp[I*t], {r, 0, 3},
{t, -Pi, Pi}, PlotStyle -> Green]
Show[{p1, p2, p3, point1},
PlotRange -> {{-3, 3}, {-3, 3},
{-3*Pi, 3*Pi}}, BoxRatios -> {1, 1, 1}]