There are several different ways to define a norm on the space of polynomials $\mathbb C [z]$. For example, $\|p\| = \sup_{|z|\le 1}|p(z)|$ defines a norm.
If $\mathbb C (z)$ denotes the field of quotients, my question is, what is the norm on $\mathbb C (z)$? I understand $\mathbb C [z]\hookrightarrow \mathbb C (z)$ so that the norm on $\mathbb C (z)$ must somehow extend the norm $\|p\|$ on $\mathbb C [z]$. My problem with that is that defining $\|p/q\| = \sup_{|z|\le 1}|p(z)/q(z)|$ does not work because $q$ can be zero.
I think there is no reasonable norm for $\mathbb C(z)$.
But maybe make it a complete metric space like this: Consider elements of $\mathbb C(z)$ to be continuous maps of the Riemann sphere to itself. Use uniform convergence (with respect to a metric for the Riemann sphere).