Meromorphic function satisfying $f(z) = f(z+1) =f(z + \sqrt{2}$) is constant.

Question

I was looking this question and answer Take $f$ and entire function such that $f(z) = f(z+1)=f(z+\sqrt{2})$, then it is constant and I wonder what would happen if $f$ is not entire but meromorphic.

Thank you.