Compute multivariate complex Gaussian integral

A corrected form of the question asks to show that $\int_{\mathbb R^n} e^{-x^tAx}\;dx\;=\; \pi^{n/2}/\sqrt{\det A}$ for symmetric $n$-by-$n$ $A$ with positive-definite real part. First, for $A$ real (positive-definite), there is a (unique) positive-definite square root $S$ of $A$, and the change of variables $x=S^{-1}y$ gives the result, as the questioner had noted.

The trick here, as in many similar situations asking for extension to complex parameters of a computation that succeeds simply by change of variables in the purely real case, is invocation of the Identity Principle from complex analysis. That is, if $f,g$ are holomorphic on a non-empty open $\Omega$ and $f(z)=g(z)$ for $z$ in some subset with an accumulation point, then $f=g$ throughout $\Omega$. This can be iterated to apply to several complex variables, in various manners. In the case at hand, this gives an extension from symmetric real matrices to symmetric complex matrices (with the constraint of positive-definiteness on the real part, for convergence of everything).

To be sure, the complex span (in the space of $n$-by-$n$ matrices) of real symmetric matrices is complex symmetric matrices, not $n$-by-$n$ complex matrices with arbitrary imaginary part.

EDIT: To discuss meromorphy in each of the entries, observe that if $A$ is symmetric with positive-definite real part, then so is $A+z\cdot (e_{ij}+e_{ji})$ for sufficiently small complex $z$, where $e_{ij}$ is the matrix with $ij$-th entry $1$ and otherwise $0$. Without attempting to describe the precise domain, this allows various proofs of holomorphy of both sides of the asserted equality. To prove connectedness of whatever that domain (for fixed $i>j$) is, it suffices to observe that it is convex: if $A$ and $B$ are symmetric complex with positive-definite real part, then the same is true of $tA+(1-t)B$ for real $t$ in the range $0\le t\le 1$.

Take \begin{align} A:= \begin{pmatrix} 1 & -2i \\ i & 1 \end{pmatrix} \end{align} Then $\sqrt{\det(A)} = \pm i$, and your formula predicts that the value of the integral is $\pm i\pi$. However, $\mathbf{x}^{T}A\mathbf{x} = x^2 + y^2 - ixy$ and \begin{align} \int_{\mathbb{R}^2} \exp(-x^2 - y^2 + ixy) \, \mathrm{d}x \mathrm{d}y = \frac{2\pi}{\sqrt{5}}. \end{align}