What happens with this way of doing it?
A rational function is a map from the Riemann sphere $\mathbb C \cup \{\infty\}$ to itself. The Riemann sphere is compact, so has a unique uniform structure. (So choose a nice metric for it, say the one from an actual sphere.) Use "uniform convergence".
This topology is metric. But not a norm: we naturally include, as one of the rational functions, the constant $\infty$.