I have complex vector $ x, y \in \mathbb{C}^ N $. I need to find a metric that depicts their similarity. When $ x, y \in \mathbb{R}^ N $, a common metric is the correlation $ \rho = \frac{x^T y}{\mid x \mid \mid y \mid} $. When dealing with complex vectors, this seems to be more tricky. I have found that sometimes $ \rho = \frac{\mid x^T y \mid}{\mid x \mid \mid y \mid} $ is used but it does not seem to work well. Are you aware of any metric that represents the similarity between two complex vectors?
Update I have also used $ \rho = \frac{\mid x^H y \mid}{\mid x \mid \mid y \mid} $ but does not seem to be fair for all cases.
What you are looking for isn't the transposition operation, but the conjugate transposition operation. Often denoted with a star, the vector $x^*$ is the complex conjugate AND the transpose of $x$. With this, the quantity
$$\rho=\frac{x^*y}{|x||y|}$$
may be what you are looking for. However, this is not symmetric with respect to $x$ and $y$, but the modulus of this quantity may also be of interest.