My question is inspired by a comment in this topic.
How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal?
One knows that $R$ is a DVR, and its field of fractions is $\mathbb C(x)$. Maybe an idea could be to use the valuation associated to $R$, or better say, the norm induced by this valuation.