There we go. I'm asking this question to know the different complex structures can be defined on $TM$ and $T^*M$ (I don't mind because my manifold will be Kähler).
I know there are related questions, for example those ones in MO (the former cites the latter) and also others in M.SE, although I can't find them. However, none of them clarify my doubts.
Suppose $M$ is a manifold. It is well known that $T^*M$ is a symplectic manifold in a natural way. It seems that doesn't happen if we are interested in complex structures, i.e., $T^*M$ isn't a (an almost) complex manifold in a natural way.
Now, suppose that $M$ is a Riemannian manifold. Róbert Szöke, in his paper Complex structures on tangent bundles of Riemannian manifolds (you should be able to find the paper here) says that there are a natural complex structure on $TM$. Namely, that the metric gives rise to a direct sum decomposition of the bundle $T(TM)$ into the vertical and the horizontal subbsendles, so for any $p\in TM$ we have the isomorphism $T_p(TM)\cong T_{\pi(p)}M\oplus T_{\pi(p)}M$.
Question 1. Im' not sure the metric give you that split. I guess that the split is actually given by the Levi-Civita connection associated with the Riemannian metric of $M$. Am I right?
Furthermore, imagine that $M$ is also complex (in my case it will be Kähler). If $J$ is the complex structure on $M$, I have checked that $TJ$ (the differential map of $J$) is an almost complex structure on $TM$. I don't know if it will be integrable, it should be though (call it poetic justice).
Question 2. I can't find the question here in M.SE, but there, there was a comment saying that if $M$ is complex, then $T^*M$ is too. Is $TJ$ the complex structure that person referred to? Or can we define (in a more or less natural way) other complex structures using $J$? In that case, does it matter if I consider $TM$ or $T*M$, suposseing $M$ is still Riemannian?
Let me add some context for my question. I want to reproduce Hitchin's result about the c-Map but defining all the structures globally. Because he points that it is possible, but he doesn't prove the result and works only locally.
Here is the paper I say and here you can find another paper studying the c-map globally in the lagange of principal bundles, which I want to avoid.
(1) You are correct that the almost complex structure $J : TTM \to TTM$ is derived from the Levi-Civita connection of the given metric. This connection can be viewed as a particular distribution $H \subset TTM$ complementary to the vertical distribution $V := \ker T\pi$, where $\pi : TM \to M$ is the canonical projection. In particular, we can define the almost complex structure on $TTM = H \oplus V$ by declaring $$J(X, Y) := (-Y, X) .$$ This also shows that actually need less information than a metric to define this almost complex structure: All we need is a distribution $H$ complementary to $V$, and we can identify any such $H$ with a linear connection $\nabla$ on $M$ and vice versa.
As Dombrowski points out in $\S$ 5 of the article mentioned below, this complex structure is integrable iff $\nabla$ it is flat and torsion-free. So, for Levi-Civita connections (and as Szöke mentions) the complex structure is integrable iff $g$ is (locally) flat, and in particular it is not integrable for nonflat Kähler metrics.
(2) For any metric, the index-raising and lowering operators define a canonical isomorphism $TM \cong T^* M$, so it's enough to ask the questions, e.g., about $TM$.