Functional equation (is there a polynomial or holomorphic solution?)

Question

I've played around with the functional equation $p(z^2+1)=p(z)^2$ for all $z \in \mathbb{C}$. Is there a non constant polynomial or holomorphic solution? Can we say anything about uniqueness? If've found a real solution which converges on all real numbers and was interested if I could extend it. The solution was $f(x):=\lim\limits_{n \to \infty}f_n(x)^{\frac{1}{2^n}} $ where $f(x):=f_1(x)=x^2+1$ and $f_n=f^{\circ n}$. I think I proved that this solution converges uniformly and is differentiable. I'd appreciate any ideas or help!