The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the several complex analysis, at wiki here: [Holomorphically convex hull][1], changing the holomorphic functions into complex linear functions.

Now my questions is what is the complex convex set looks like.

First, the complex convex set must be convex set, but does every convex set must be complex convex?

If not, at least in the one complex dimension case, the complex convex is complex polynormally convex, if it is a compact, its complement must be connected. Can you say something more?

   The same question is at [here][2].


  [1]: http://en.wikipedia.org/wiki/Holomorphically_convex_hull
  [2]: https://math.stackexchange.com/questions/356895/how-does-the-complex-convex-set-look-like