An open or compact set in $\overline \mathbbb{C}$ is called to be convex in the positive direction of real axes iff its intersection with any closed right halphplain is acyclic.

There are known natural properties.

The complementary set $\overline \mathbbb{C}\setminus M$ is convex in oposite directions.

A set in $\mathbbb C $ is convex iff it is convex in all directions.