In a real inner-product space, $x\perp y$ iff $\|x+y\|^2=\|x\|^2+\|y\|^2$, since the difference is $2\langle x,y\rangle$. Over $\mathbb C$, that equality implies only $\operatorname{Re}\langle x,y\rangle=0$, not orthogonality. For example $x=1,y=i$ satisfies the length equality but $\langle x,y\rangle=i\ne0$.
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