Every exotic $\mathbb{R}^4$ is parallelisable (as $\mathbb{R}^4$ is contractible), and therefore admits an almost complex structure. It is a result of Gromov that an open manifold of dimension six or less admits a complex structure if and only if it admits an almost complex structure. Therefore every exotic $\mathbb{R}^4$ admits a complex structure.

It is a result of Gromov that an open manifold of dimension six or less admits a complex structure if and only if it admits an almost complex structure; see the corollary on page 103 of his book Partial Differential Relations. As $\mathbb{R}^4$ is contractible, every exotic $\mathbb{R}^4$ is parallelisable. Therefore, they all admit almost complex structures, and hence complex structures by Gromov's result.