The standard notation for a Hermitian metric looks like this: $$\sum ds^2 = \sum dz_i \otimes d\overline{z_j}.$$
The conjugate confused me for a while until I explained it to myself as follows. In tensors, we must have
$$w dz_i \otimes d\overline{z_j} = (w\, dz_i) \otimes d\overline{z_j} =dz_i \otimes (w\, d\overline{z_j}).$$
So if we define the hermitian product at a point to be $$h(v,u)=g(v,\overline u),$$ where $g$ is the think we get by applying the tensor to the holomorphic vector and anti-holomorphic vector, then we get anti-linearity in the second argument, as desired.
However, this explanation doesn't work in the following situation.
Suppose we have a metric that looks like $$h^2 dz_1 \otimes d\overline{z_1}.$$ Define the coframe $\phi = h \, dz_1$. Griffiths and Harris say (p. 77) we can rewrite this as $$\phi \otimes \overline{\phi}.$$ This looks like $$h \, dz_1 \otimes \overline {h\, dz_1}=|h|^2 dz_1 \otimes d\overline{z_1},$$ which is not what it should be.
How should I understand this notation so that everything works?
We have $h=|h|$ because $h$ is positive. This is forced by the positive definite hypothesis on the inner product.