Let $a,r\in\Bbb R^*_+$.
I choose as definition of complex logarithm, the one defined by $\log(z)=\log(|z|)+i\theta$ with $z=|z|e^{i\theta}$ et $\theta\in\,]-\pi,\pi[$.
So $(ir)^{ia}=e^{ia\ln(ir)}=e^{ia(\ln(r)+i{\pi\over2}})=e^{ia(\ln(r)}e^{-a\pi\over2}$. And hence the modulus of $(ir)^{ia}$ is $e^{-a\pi\over2}$. Is it true?
On the other hand if $W_{\lambda,\mu}(z)\sim\mathrm{e}^{-z/2}z^\lambda$
as $|z|\to\infty$. We have
$$W_{ia,\mu}(ir)\sim\mathrm{e}^{-ir/2}(ir)^{ia}$$?
Can we say $$|W_{ia,\mu}(ir)|\sim e^{-a\pi\over2}$$