It's possible to construct complex periodic functions with two periods in different directions, such as $f(z) = \cos x + i \sin 2y$. That has periods $2\pi$ and $\pi i$. It's also not analytic.
It's been a long time since complex variables, and that was self-study, so I'm very likely under-thinking this, but...Is there any analytic function with two linearly-independent periods?
I don't consider constant functions as properly periodic, since there's no minimum period...but I'm not sure if that attitude is mainstream.