Bounds on complex Hermite polynomials

Question

There is a well-known bound on Hermite polynomials: $$ \left|\cfrac{H_n(x)}{\sqrt{2^n n!}}\right|\le e^{0.5x^2}, $$ where $x$ is real. I am trying to find a bound of the following form: $$ \left|\cfrac{H_n(ix)}{\sqrt{2^n n!}}\right|\le e^{\lambda n}f(x), $$ for the case of imaginary arguments, where $f(x)$ does not depend on $n$. I would be very grateful for suggestions. Numerical analysis indicates that such bounds might be possible. However, all bounds that I found in the literature are for real arguments.