Two-Dimensional Gaussian integral in complex coordinates.

It follows from my Phys.SE answer here that OP's Gaussian integral becomes $$ \begin{align} I~:=~&\int_{\mathbb{R}^2} \! \frac{\mathrm{d}{\rm Re}\alpha \wedge \mathrm{d}{\rm Im}\alpha}{\pi}~ \exp\left\{-\mu \alpha^{2}-\nu \alpha^{* 2}-z^{*} \alpha+z \alpha^{*}-|\alpha|^{2}\right\}\cr ~=~&\int_{\mathbb{C}} \! \frac{\mathrm{d}\alpha^{\ast} \wedge \mathrm{d}\alpha}{2\pi i}~ \exp\left\{ -\frac{1}{2}A^TSA +Z^TA\right\}\cr ~=~&\sqrt{\frac{-1}{\det(S)}}\exp\left\{\frac{1}{2}Z^TS^{-1}Z \right\}\cr ~=~&\frac{1}{\tau} \exp \left\{-\frac{\mu z^2+\nu z^{* 2}+z z^{*}}{\tau^2}\right\}. \end{align}$$

Here we have defined $$ \begin{align} A~:=~&\begin{pmatrix} \alpha \cr \alpha^{\ast} \end{pmatrix}, \cr Z~:=~&\begin{pmatrix} -z^{\ast} \cr z \end{pmatrix}, \cr J~:=~&\begin{pmatrix} 1 & i \cr 1 & -i \end{pmatrix}, \cr S~:=~&\begin{pmatrix} 2\mu & 1 \cr 1 & 2\nu \end{pmatrix}, \cr \tau^2~:=~&-\det(S), \cr S^{-1}~=~&\frac{1}{\tau^2}\begin{pmatrix} -2\nu & 1 \cr 1 & -2\mu \end{pmatrix}, \end{align}$$

Moreover, the Gaussian integral $I$ is convergent if ${\rm Re}(J^TSJ)>0$ is positive definite.