Is it possible to 'find' all metrics on $\mathbb{C}[x,y]$?
If not, at least:
Is there a metric on $\mathbb{C}[x,y]$ such that the sub-algebra $\mathbb{C}[p,q]$, where $p,q \in \mathbb{C}[x,y]$ satisfy $p_xq_y-p_yq_x \in \mathbb{C}^*$, is dense in $\mathbb{C}[x,y]$?
How much this depends on metrics on $\mathbb{C}$? (if at all).
This question is relevant.