Michael Trott in his book The Mathematica GuideBook for Symbolics, p. 1004, has illustrated a nice way to visualize the Riemann surface of the incomplete gamma function $\Gamma(\alpha, z)$. To plot the Riemann surface of, for example, $\Gamma\left(\frac{3+i}{10}, z\right)$, we have:
With[{α = 0.3 + 0.1 I, ε = 10^-12},
GraphicsRow[Graphics3D[{EdgeForm[Thickness[0.002]],
SurfaceColor[Hue[0.09], Hue[0.18], 2.3],
Table[Last /@ Partition[Cases[
ParametricPlot3D[{r Cos[φ], r Sin[φ],
#[Exp[2 k π I α] Gamma[α, r Exp[I φ]] +
(1 - Exp[2 k π I α]) Gamma[α]]},
{r, 0, 2}, {φ, -π + ε, π - ε},
PlotPoints -> {30, 40}], _Polygon, ∞], 2],
{k, -2, 2}]}, BoxRatios -> {1, 1, 2.5},
PlotRange -> All] & /@ {Re, Im}]]
where we have used the identity:
$$\Gamma\left(\alpha, \exp(2 k \pi i)z\right) = \exp(2 k \pi i \alpha)\, \Gamma\left(\alpha, z\right) + (1- \exp(2 k \pi i \alpha)) \Gamma\left(\alpha\right)$$
However, Mathematica does not return anything.
Any help is appreciated!