Prove that an analytic function is zero

Question

Suppose that I have an analytic function in four variables $$f = f(z_1,z_2,z_3,z_4)$$ such that I know the following facts: $$\Re [f(z_1,z_2,0,0)] = 0$$ and that: $$f(0,0,z_2,z_3) = 0$$ I was wondering whether these constraints are enough to say that the function is zero everywhere in $\mathbb{C}^4$ or not. I would be interested in knowing which tools one might use in understanding this.

Answer 1

Here is a counterexample $$f(z_1, z_2, z_3, z_4) = z_1 z_2 z_3 z_4$$