Existence of non-trivial complex unit neighborhoods

First, we briefly mention the basic definitions and results on unit neighborhoods of zero. Next, we show the existence of certain non-trivial complex unit neighborhoods of zero. We expose a generalization of the construction method used on the mentioned particular case. Since this construction may not lead to a unit neighborhood of zero, we develop some necessary conditions. Finally, we describe our heuristic use of Wolfram Mathematica to prove the existence of non-trivial complex unit neighborhoods.