Functional analysis. What are the almost periodic functions in the complex plane?

The almost periodic functions in the real line can be characterized as uniform limits of trigonometric functions. I was wondering if there is a similar definition in the complex plane (a locally compact group under addition).

In particular, I'm trying to find out if there is a quasi-periodic, non-constant function $ f $ in $ mathbb {C} $ such that $ f $ it is invariant under rotations that is, $ f (tz) = f (z) $ for all $ t in mathbb {T} $, $ z in mathbb {C} $.

Any help is really appreciated.