I have seen talks about this, though somewhat over my head. But, yes absolutely, people do research such things. A. Katok is one name that comes to mind here. See this paper: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.2082
They consider actions of $\mathbb{Z}^k$ and $\mathbb{R}^k$ that generalize the usual time-like case of $\mathbb{Z}$ and $\mathbb{R}$.
This is not my expertise, though, so I can't say what this says about the $\mathbb{C}$ case you're interested in. But at least I can say, yes, there is work in the higher-dimensional case.