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][1] 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 $\mathbb C^2$. [1]: http://www.numdam.org/article/SB_1964-1966__9__7_0.pdf

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 $\mathbb C^2$.