So if we have a two dimensional Gaussian function $\frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}$, then the following integration $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{1}{2\pi}e^{-\frac{x^2+y^2}{2}}dxdy=1$$
Now if we switch to complex variables $z,z^*$, what should be the range of those complex variables for the same integration above transformed into complex variable?
$$\int\int\frac{i}{2}\frac{1}{2\pi}e^{-\frac{|z|^2}{2}}dzdz^*=1$$