Integrability of an almost complex structure vs holomorphicity of the section $M\rightarrow \mathcal{J}(M)$ Let's say we have an almost complex manifold $(M, J)$. Consider the complex vector bundle $V\rightarrow M$ whose fiber over $x$ is the space of almost complex structures on $T_x M$. 
Is there any logical connection between the following two conditions:


*

*$J$ is integrable;

*the map $M\rightarrow V$ associated to $J$ is holomorphic?

 A: First of all, there is no reasonable complex structure on the total space of this bundle — surely, it should be the usual complex structure on the space $\mathrm{SO}(2n)/\mathrm{U}(n)$ along the fibres; and it would be natural to extend it tautologically in the horizontal direction, i. e. for $(\iota,x)\in\mathcal{J}(M)$ a complex structure operator on the tangent space $T_xM$, one would have $I(v) = \widetilde{\iota(d\pi(v))}$, where $\pi \colon \mathcal{J}(M) \to M$ is the projection and $v \in T_{(\iota, x)}\mathcal{J}(M)$ is a horizontal vector. The issue is that there is no natural notion of a “horizontal vector” unless you pick up a connection in the bundle $\mathrm{End}(TM)$. However, if your section is horizontal w. r. t. a connection in the bundle of endomorphisms which comes from a torsion free connection in the tangent bundle $TM$, then your almost complex structure would be integrable according to a well-known theorem in differential geometry, which is discussed e. g. in the great MO post on the geometric meaning of torsion.
A: This is an answer to what was perhaps the intended question (taking into account the comment to the answer by @gulag57). Let $U$ be an open set in $\mathbb{C}^n$. An almost complex structure on $U$ can be regarded as a map $J \colon U \to \text{GL}(2n;\mathbb R)/\text{GL}(n;\mathbb C)$. The target space has a canonical structure of complex manifold, as is elegantly explained in section 1 of this Bourbaki seminar by Douady. One may then ask whether complex structures on $U$ correspond to maps $J$ which are holomorphic. That is not the case.
Almost complex structures which are sufficiently close to the standard complex structure on $U$ are parametrized by maps $\theta \colon U \to \text{Hom}( \overline{ \mathbb C^n}, \mathbb C^n)$ (there are canonical complex charts around each point in $\text{GL}(2n;\mathbb R)/\text{GL}(n;\mathbb C)$ taking values in $\text{Hom}( \overline{ \mathbb C^n}, \mathbb C^n)$ as is also explained in Douady's paper). The map  $\theta$ will correspond to an integrable structure if and only if it satisfies the Maurer-Cartan equation $\overline{\partial} \theta + [\theta,\theta]=0$ (see for instance "Complex Geometry" by Daniel Huybrechts, Lemma 6.1.2 p. 258). This is very different from the map $\theta$ being holomorphic.
For instance, the anti-holomorphic map defined by
$$\theta(z_1,z_2) = \overline{z_2} d\overline{z_1} \otimes \frac{\partial}{\partial z_1} + \overline{z_1} d\overline{z_2} \otimes \frac{\partial}{\partial z_1}$$
or, in matrix notation,
$$ \theta(z_1,z_2) = \left[ \begin{array}{cc} \overline{z_2} & \overline{z_1} \\ 0 & 0 \end{array} \right] $$
satisfies the Maurer-Cartan equation and therefore corresponds to a complex structure on $\mathbb C^2$.
