Expressing a "generalisation" of Gamma function in terms of known special functions.

Question

It is known that an integral of the form $$\int_0^\infty e^{-x^2} x^\beta \, dx $$ for $\beta > -1$ can be expressed in terms of a Gamma function. I am interested in the integral $$I(\alpha,\beta) = \int_0^\infty e^{i \alpha x -x^2} x^\beta \, dx $$ and whether it can be expressed in terms of any known special functions. Here $\alpha \in \mathbb{C}$. Of course, if $\beta$ was a positive integer one could express $I(\alpha,\beta)$ in terms of derivatives an error function. This is because you could write $$I(\alpha,\beta) = (-i)^\beta \frac{d^\beta}{d\alpha^\beta}\int_0^\infty e^{i \alpha x -x^2} \, dx$$ and $\int_0^\infty e^{i \alpha x -x^2} \, dx$ can be written as a (complex) error function. However it is the case of non-integer $\beta$ that interests me.