I am trying to explicitly find the relationship between a Hermitian metric on a complex manifold and the Riemannian metric on the underlying real manifold -- and specifically on how the determinants of both are related. Sorry for the length of what follows.
Consider a metric on a (patch of) an even dimensional real manifold $\mathcal{M}$ of dimension $2n$, where the Riemannian metric is represented as the $(0,2)$ tensor $G_{MN}$, with $M, N = 1, \dots, 2n$, and coordinates labelled $X^M$. Letting $\mu, \nu = 1, \dots, n$, with
\begin{eqnarray} \label{eq:G-def} G_{MN} = \begin{pmatrix} g_{\mu\nu}^{(1)} & \vert & g_{\mu\nu}^{(2)} \\ \hline g_{\mu\nu}^{(3)} & \vert & g_{\mu\nu}^{(4)} \end{pmatrix} \end{eqnarray}
$g_{\mu \nu}^{(2)} = g_{\nu \mu}^{(3)}$, $g_{\mu \nu}^{(1)} = g_{\nu \mu}^{(1)}$, $g_{\mu \nu}^{(4)} = g_{\nu \mu}^{(4)}$.
Let $X^M = (x^{\mu}, y^{\nu})$, giving us bases $\{\dfrac{\partial}{\partial x^{\mu}}, \dfrac{\partial}{\partial y^{\nu}} \}$ for $T_p\mathcal{M}$, $\{ dx^{\mu}, dy^{\nu}\}$ for $T_p^*\mathcal{M}$, and suppose that $\mathcal{M}$ has an integrable almost complex structure $J$.
Let $z^{\mu} = x^{\mu} + i y^{\mu}$, $\bar{z}^{\mu} = x^{\mu} - i y^{\mu}$, and with the $dz^{\mu}$, $d\bar{z}^{\mu}$, $\frac{\partial}{\partial z^{\mu}}$, $\frac{\partial}{\partial \bar{z}^{\mu}}$ defined as above.
\begin{eqnarray} \label{metric-complexified} ds^2 &=& G_{MN} dX^M \otimes dX^N \\ % &=& g_{\mu \nu}^{(1)} dx^{\mu} \otimes dx^{\nu} + g_{\mu \nu}^{(2)} dx^{\mu} \otimes dy^{\nu} + g_{\mu \nu}^{(3)} dy^{\mu} \otimes dx^{\nu} + g_{\mu \nu}^{(4)} dy^{\mu} \otimes dy^{\nu} % \\ % &=& \dfrac{1}{4}\left(g_{\mu \nu}^{(1)} - ig_{\mu \nu}^{(2)} - ig_{\mu \nu}^{(3)} - g_{\mu \nu}^{(4)}\right)(dz^{\mu} \otimes dz^{\nu}) + \dfrac{1}{4}\left(g_{\mu \nu}^{(1)} + ig_{\mu \nu}^{(2)} + ig_{\mu \nu}^{(3)} - g_{\mu \nu}^{(4)}\right)(d\bar{z}^{\mu} \otimes d\bar{z}^{\nu}) \\ && + \dfrac{1}{4}\left(g_{\mu \nu}^{(1)} + ig_{\mu \nu}^{(2)} - ig_{\mu \nu}^{(3)} + g_{\mu \nu}^{(4)}\right)(dz^{\mu} \otimes d\bar{z}^{\nu}) + \dfrac{1}{4}\left(g_{\mu \nu}^{(1)} - ig_{\mu \nu}^{(2)} + ig_{\mu \nu}^{(3)} + g_{\mu \nu}^{(4)}\right)(d\bar{z}^{\mu} \otimes dz^{\nu}) \\ % &=& g_{\mu \nu}(dz^{\mu} \otimes dz^{\nu}) + g_{\bar{\mu} \bar{\nu}}(d\bar{z}^{\mu} \otimes d\bar{z}^{\nu})+ g_{\mu \bar{\nu}}(dz^{\mu} \otimes d\bar{z}^{\nu}) + g_{\bar{\mu} \nu}(d\bar{z}^{\mu} \otimes dz^{\nu}) \end{eqnarray}
(I suppose this can be described as the $\mathbb{C}$-extension of the original metric.)
A Riemannian metric $g$ is Hermitian if for any $X,Y \in T_p\mathcal{M}$, $p\in \mathcal{M}$, we have
$g(J(X),J(Y)) = g(X, Y) $.
We assume that the almost complex structure acts as $J\left(\frac{\partial}{\partial x^{\mu}}\right) = \frac{\partial}{\partial y^{\mu}}$, $J\left( \frac{\partial}{\partial y^{\mu}}\right) = -\frac{\partial}{\partial x^{\mu}}$, so that
$ J\left(\dfrac{\partial }{\partial z^{\mu}} \right) = i \dfrac{\partial}{\partial z^{\mu}} \,,\quad J\left(\dfrac{\partial}{\partial \bar{z}^{\mu}}\right) = -i \dfrac{\partial}{\partial \bar{z}^{\mu}} $
The above condition then forces $g_{\mu \nu} = g_{\bar{\mu}\bar{\nu}} = 0$.
Looking at the metric expansion, we see that this means $g_{\mu \nu}^{(1)} = g_{\mu \nu}^{(4)}$, while $g_{\mu \nu}^{(2)} = - g_{\mu \nu}^{(3)}$, in this basis.
Using these results, we find that
\begin{eqnarray} G_{MN} \, dX^M \otimes dX^N &=& \frac{1}{4}\left(g_{\mu \nu}^{(1)} + g_{\mu \nu}^{(4)} + i\left(g_{\mu \nu}^{(2)} - g_{\mu \nu}^{(3)} \right)\right) dz^{\mu} \otimes d\bar{z}^{\nu} \\ && + \frac{1}{4}\left(g_{\mu \nu}^{(1)} + g_{\mu \nu}^{(4)} - i\left(g_{\mu \nu}^{(2)} - g_{\mu \nu}^{(3)} \right)\right) d\bar{z}^{\mu} \otimes dz^{\nu} % \\ % &=& \frac{1}{2}\left(g_{\mu \nu}^{(1)} + ig_{\mu \nu}^{(2)}\right) dz^{\mu} \otimes d\bar{z}^{\nu} + \frac{1}{2}\left(g_{\mu \nu}^{(1)} - ig_{\mu \nu}^{(2)}\right) d\bar{z}^{\mu} \otimes dz^{\nu} % \\ % &=& \frac{1}{2} \left( h_{\mu \bar{\nu}}\, dz^{\mu} \otimes d\bar{z}^{\nu} + h_{\bar{\nu} \mu} \, d\bar{z}^{\nu} \otimes dz^{\mu} \right) \end{eqnarray}
Where we call $h_{\mu \bar{\nu}} = \overline{h_{\bar{\nu}\mu}}$ the Hermitian metric on $\mathcal{M}$.
QUESTION Is the above relationship between $h_{\mu \bar{\nu}}$ and $G_{MN}$ correct, and why is it that $\det(G_{MN}) = (\det(h_{\mu \bar{\nu}}))^2$?
Here is an attempt at an answer. Let $A$, $B$, $C$, $D$ be $n \times n$ matrices. Then if $C$, $D$ commute, we have the identity
\begin{equation} \label{det:identity} \det \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det(AD - BC) \end{equation}
Using the decomposition of $G_{MN}$, we find that
\begin{eqnarray} \label{det-G} \det\left(G_{MN}\right) &=& \det\left(g_{\mu\nu}^{(1)}g_{\nu\lambda}^{(4)} - g_{\mu\nu}^{2}g_{\nu\lambda}^{3}\right) = \det\left( \left(g_{\mu\nu}^{(1)}\right)^2 + \left(g_{\mu\nu}^{(2)}\right)^2\right) \\ &=& \det\left( g_{\mu\nu}^{(1)} + i g_{\mu\nu}^{(2)}\right) \det\left( g_{\mu\nu}^{(1)} - i g_{\mu\nu}^{(2)}\right) \end{eqnarray}
The second equality above comes from the identifications due to hermitian condition described above.
The RHS of the last equation is $\det\left(h_{\mu\bar{\nu}}\right) \det\left(h_{\bar{\nu}\mu}\right)$ and we have seen that $h_{\bar{\nu}\mu}=\overline{h_{\mu \bar{\nu}}}$.
So we find that
$$ \det\left(G_{MN}\right) = \left|\det\left(h_{\mu\bar{\nu}}\right)\right|^2 $$
or $\det\left(h_{\mu \bar{\nu}}\right) = \sqrt{\det\left(G_{MN}\right)}$ if $h_{\mu \bar{\nu}}$ is real.
Problems: $g^{(2)}$, $g^{(3)}$ do not necessarily commute. But perhaps it is true that these components are somehow required to vanish -- maybe due to the requirement that $\{ \frac{\partial}{\partial x^{\mu}}, \frac{\partial}{\partial y^{\nu}} \}$ form an /orthonormal basis/ -- so that $h_{\mu \bar{\nu}} = g_{\mu \nu}^{(1)}$.