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

In particular, I am trying to figure out if there exists a non-constant almost periodic function $f$ on $\mathbb{C}$ such that $f$ is invariant under rotations i.e. $f(tz) = f(z)$ for all $t\in \mathbb{T}$, $z\in \mathbb{C}$.

Any help is much appreciated.