Explicit determination of the open book in $S^3$

Question

I'm familiar with the fundamental concepts of algebraic and differential topology.

How can I determine explicitly the topology of a page of the open book in $S^3$ given by for example $$f: \mathbb{C}^2 \to \mathbb{C}, \ f(z_1, z_2) = z_1z_2.$$

Any reference request or suggestion will be appreciated.