I've seen people write $g(z) = dzd\bar{z}$ to refer to the standard conformal metric on the complex plane $\mathbb{C}$ (e.g., on Wikipedia). I thought this was referring to the tensor product $dz \otimes d\bar{z}$, but $dz \otimes d\overline{z}$ is a Hermitian metric and not a Riemannian metric (in particular, it is not real-valued). The Wikipedia page linked earlier defines a conformal metric as a Riemannian metric. Am I missing something?
Edit: To clarify, here is what I was thinking when I said $dz \otimes d\bar{z}$ is Hermitian: $$ (dx + idy) \otimes (dx - idy) = dx \otimes dx + dy \otimes dy + i(dy \otimes dx - dx \otimes dy). $$ As tensors, $dy \otimes dx$ and $dx \otimes dy$ are distinct, e.g. $(dy \otimes dx)(\partial / \partial y, \partial / \partial x) = 1$ but $(dx \otimes dy)(\partial / \partial y, \partial / \partial x) = 0$.