**Edit:** According to the answers to the linked MSE question and the comment of Holonomia, I understand that the answer to the second question is that **" Every tangent bundles is a complex manifold"**.

Let $M$ be a compact real manifold. Assume that the tangent bundle $TM$ carries a holomorphic structure, namely it is equipped with a holomorphic atlas. We fix a holomorphic structure for $TM$.

We say that a map $f:M \to M$ has a holomorphic derivative if $Df:TM \to TM$ is a holomorphic map.

The first Question:Is it true to say that the space of all diffeomorphism of $M$ with holomorphic derivative admits a structure of a finite dimensional Lie group?

The Second Question:What is an example of a manifold whose tangent bundle, as a manifold, does not admit a holomorphic atlas?

I asked the latter question on MSE but I did not get any answer.

https://math.stackexchange.com/questions/2611104/some-questions-on-the-tangent-bundle-of-manifolds

# Further questions:

Added: How can one decide that a given map has holomorphic derivative? In this investigation, and motivated by CR equations, what kind of differential operators would appear?

As a particular example, we consider the Hopf map $p: S^3 \to S^2$. Is it a map with holomorphic derivative? Of course this question is meaningless if we do not fix a holomorphic structure for $TS^2$ and $TS^3$. So it is natural to ask: What is a precise holomorphic structure for these space? Can the holomorphic structure of the tangent bundle of a Riemann surface or a parallelizable manifold be determined explicitly?