The bilateral Laplace transform of a Gaussian function could be established as: $$e^{x^2/2}=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}e^{-xy}e^{-y^2/2} dy$$
Then what should be a similar relation for a Gaussian function with complex variable:$e^{z^2/2}$?