Diffeomorphisms on a real manifold whose derivative are holomorphic maps on the tangent bundle 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?
 A: I guess that you mean: $TM$ carries an almost complex structure $J:TM\to TM$ with $J^2=-1$. It is integrable to complex structure iff the Froelicher-Nijenhuis bracket $[J,J]$ vanishes.
First question: The Lie algebra of your group consists of all vector fields $X$ with $\mathcal L_XJ =  [X,J]=0$. Since $J$ is invertible, $[X,J]=0$ is an elliptic equation of 1. order for $X$, so the solution space is finite dimensional on a compact manifold.
Second question: If $J$ exists, the manifold must be even dimensional and orientable.
Added:
So let us assume that $TM$ carries a complex structure. $TM$ is not compact, so the group of biholomorphic automorphismsm need not be finite dimensional.
First question: $\phi\mapsto T\phi=D\phi$ is an injective group homomorphism into the group of all those biholomophic diffeomorphisms of $TM$ which are also vector bundle homomorphisms; i.e., linear between fibers. Its Lie algebra consists of all vector fields which are holomorphic and linear when restricted to any fiber. To be holomorphic is an elliptic equation, and the linearity along fibers should imply that the corresponding solution space is finite dimensional.
