Let $(M,J)$ be a complex manifold and let $z^i=x^i+iy^i,\ i=1,\ldots,{\rm dim_\mathbb C \ M}$ be its coordinates on a local patch.
Here J is the complex structure, which, for all $p\in M$, is a map $J:\ T_pM\longrightarrow T_pM$ satisfying $J_p=-id_{T_pM}.$
In particular it can be extended to the complexified tangent space $T_pM^{\mathbb C}=T_pM\otimes \mathbb C$.
A complex manifold always admits a hermitian metric, satisfying $g(JX,JY)=g(X,Y), \ \forall X,Y\in T_pM$. Again this tensor can be extended to the complexified tangent space.
Once this metric is considered, we induce a connection and one can compute the torsion $T$ or the Riemann curvature tensor $R$, which again can be extended to $T_pM^{\mathbb C}$.
Why do we consider the extension of all these tensors to $T_pM^{\mathbb C}$?
Why should we be interested in computing, for example, the Ricci curvature $S\left(\frac{\partial}{\partial z^i},\frac{\partial}{\partial \bar z^{j}} \right)\equiv S_{i\bar j}$ or the Ricci scalar?
Does the extension of these maps to the complexified vector spaces tell something about the manifold $M$?
My questions stems from the fact that the complexified tangent space has double the dimension of the true tangent space, therefore it is not clear to me what we are actually computing when we are dealing with the former.