We normalize the wave function to $1$, but couldn't we also normalize it to $-i$ as $(-i)^2=1$?
Does this not work? Is it equivalent?
Normalizing $\psi$ to $1$ means that we ensure that
$$ \int|\psi|^2dx = 1 $$
normalizing it to $-i$ would presumably mean ensuring that
$$ \int|\psi|^2dx = -i $$
which is impossible because the integrand $|\psi|^2$ is positive everywhere.