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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.