In Ballmann's book p.23, there is
Let $M$ be a complex manifold with corresponding complex structure $J$. We say that a Riemann metric $g$ is compatible with $J$ if $$g(X,Y)=g(JX,JY)$$ for all vector fields $X,Y$ on $M$.
In case $M$ is the complex plane $\mathbb C$, we have $J(\frac{\partial}{\partial x})=\frac{\partial}{\partial y}$ and $J(\frac{\partial}{\partial y})=-\frac{\partial}{\partial x}$, so $J$ can be treated as a rotation by 90 degrees, and the geometric meaning of $g(X,Y)=g(JX,JY)$ is the metric $g$ is invariant under the rotation, but I still wonders why the Hermitian metric should require the metric being invariant under this kind of rotation, does there exists a more essencial explanation?
Compare with the case of inner products on complex vector spaces. For a Hermitian inner product on a complex vector space, the real part of course satisfies $\Re(\langle iz,iw\rangle)=\Re(\langle z,w\rangle)$. Moreover, the original Hermitian inner product can be recovered from its real part, since $\langle z,w\rangle=\Re(\langle z,w\rangle)+i\Re(\langle z,iw\rangle)$. Conversely, if you have a real inner product $b$ on your vector space such that $b(iz,iw)=b(z,w)$ then you can extend it to a Hermitian inner product by $(z,w)\mapsto b(z,w)+ib(z,iw)$. A Hermitian inner product therefore is equivalent to a real inner product that is compatible with multiplication by $i$ and this can be transferred point-wise to manifolds.