= Solution
A <complex power> requires a choice of <complex logarithm>. Taking the <principal argument> of $-e$ to be $+\pi$ gives $\operatorname{Log}(-e)=1+i\pi$, hence
$$
\boxed{(-e)^{i\pi/2}=\exp\left(\frac{i\pi}{2}(1+i\pi)\right)=i e^{-\pi^2/2}.}
$$
Thus $a=0$ and $b=e^{-\pi^2/2}$ for this principal value. If the power is interpreted as multivalued, the logarithms are $1+i(2k+1)\pi$ and the possible values are $i\exp[-(2k+1)\pi^2/2]$, $k\in\mathbb Z$. Applying an identity for real powers without specifying this branch would lose the ambiguity.
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