  For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free
$k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$
is monochromatic. We study the complexity of deciding for fixed $F$ and $k$
whether a given simple digraph admits an $F$-free $k$-coloring. Our main focus
is on the restriction of the problem to planar input digraphs, where it is only
interesting to study the cases $k \in \{2,3\}$. From known results it follows
that for every fixed digraph $F$ whose underlying graph is not a forest, every
planar digraph $D$ admits an $F$-free $2$-coloring, and that for every fixed
digraph $F$ with $\Delta(F) \ge 3$, every oriented planar graph $D$ admits an
$F$-free $3$-coloring.
  We show in contrast, that
  - if $F$ is an orientation of a path of length at least $2$, then it is
NP-hard to decide whether an acyclic and planar input digraph $D$ admits an
$F$-free $2$-coloring.
  - if $F$ is an orientation of a path of length at least $1$, then it is
NP-hard to decide whether an acyclic and planar input digraph $D$ admits an
$F$-free $3$-coloring.
