I will address the case of $C$-action (as time). In the case when the phase space is also $C$, you are right, there are only Moebius transformations, which can be loxodromic (elliptic, hyperbolic) or parabolic. That is up to conjugacy by an arbitrary Moebius transformation you have $z\mapsto e^{\alpha t}z$ and $z\mapsto z+t$, where $t$ is your complex "time".

In the case when the phase space is $C^n$, we have a lot of possibilities. There is a hudge (infinite-dimensional) continuous group of analytic bijective transformations of $C^n$ onto itself. You can take any complex 1-parametric subgroup of this group. For $n=2$, there is a very nice paper of Milnor on this group.

I will address the case of $C$-action (as time). In the case when the phase space is also $C$, you are right, there are only Moebius transformations, which can be loxodromic (elliptic, hyperbolic) or parabolic. That is up to conjugacy by an arbitrary Moebius transformation you have $z\mapsto e^{\alpha t}z$ and $z\mapsto z+t$, where $t$ is your complex "time".

In the case when the phase space is $C^n$, we have a lot of possibilities. There is a hudge (infinite-dimensional) continuous group of analytic bijective transformations of $C^n$ onto itself. You can take any complex 1-parametric subgroup of this group. For $n=2$, there is a very nice paper of Milnor on this group.

Edit: Friedland, Shmuel; Milnor, John Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dynam. Systems 9 (1989), no. 1, 67–99.