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:
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.
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 $\mathbb C^2$.
