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A majority coloring of a digraph $D$ with $k$ colors is an assignment $\pi :V\left(D\right)\to \{1,2,\cdots ,k\}$ such that for every $v\in V\left(D\right)$ we have $\pi \left(w\right)=\pi \left(v\right)$ for at most half of all out-neighbors $w\in N^{+}\left(v\right)$. A digraph $D$ is majority $k$-choosable if for any assignment of lists of colors of size $k$ to the vertices, there is a majority coloring of $D$ from these lists. We prove that if $U\left(D\right)$ is a 1-planar graph without a 4-cycle, then $D$ is majority 3-choosable. And we also prove that every NIC-planar digraph is majority 3-choosable.