The incomplete gamma function is defined as follows:
$$ \Gamma(s,z)=\int_z^{\infty}t^s e^{-t}\frac{dt}{t}, \qquad Re(s)>0, \quad z\in \mathbb C $$
Since $t^s$ is a multi-valued function, this function must be multi-valued. In particular, for $z=-2$, it can be different path to $\infty$. For example, $z=-2+iy$ and $z=-2-iy$, $y\geq 0$.
What is the standard choice of the path, and of the branch cut? Mathematica calculate the value of $\Gamma(3/2,-2) \approx 0.886227 + 7.10586i$. Which choice coincide with this result?