Is the converse of the Pythagorean Theorem false for complex inner products?

You are right.

And it has to be that way, because the norm that the Hermitian product induces on $\mathbb C^n$ is the same as the Euclidean norm on $\mathbb R^{2n}$.

Therefore, for a fixed $x\in\mathbb C^n$, the set $$ \{ y\in\mathbb C^n \mid \|x+y\|^2 = \|x\|^2+\|y\|^2 \} $$ has dimension $2n-1$ over $\mathbb R$ -- and therefore it cannot be a linear subspace over $\mathbb C$. In particular it cannot equal $\{y\in\mathbb C^n \mid \langle x,y\rangle = 0 \}$.