In a real inner product space, two vectors are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$. Similarly, $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}) = \dfrac{\langle\mathbf{u},\mathbf{v}\rangle}{\langle\mathbf{u},\mathbf{u}\rangle} \mathbf{u}.$$
My question is, are the below definitions in a complex inner product space? $$\operatorname{Re}\langle\mathbf{u},\mathbf{v}\rangle = 0 \iff\mathbf{u}\perp\mathbf{v},\qquad \qquad \operatorname{proj}_{\mathbf{u}}(\mathbf{v}) = \frac{\operatorname{Re}\langle\mathbf{u},\mathbf{v}\rangle}{\langle\mathbf{u},\mathbf{u}\rangle} \mathbf{u}$$
This is false even in one dimension. Let $V = \mathbb{C}$ as a complex vector space, $u = 1$, $v = -i$. Then $\langle u,v\rangle = 1\cdot\overline{-i} = i$, so $\text{Re}\langle u,v\rangle = 0$, yet $\langle u,v \rangle \neq 0$, i.e. $u\not\perp v$. The statement about the projection also fails for this choice of $u$ and $v$, as the projection would be zero, and in one complex dimension a zero projection is possible only if either $u$ or $v$ is zero, since any two complex numbers are complex linearly dependent.