Let (for ) be the exponential map. Note that for all real numbers and goes to really fast: the dynamics of on is trivial. But the dynamics of on is completely different. First note that : the map is not a diffeomorphism, but a covering map branching at the origin. The following theorem was conjectured by Fatou (1926) and proved by Misiurewicz (1981).
So the exponential map is chaotic on the complex plane.
The exponential map is chaotic: An invitation to transcendental dynamics,
Zhaiming Shen and Lasse Rempe-Gillen arXiv