It is widely known that on a finite-dimensional vector space over a complete field, every norm is equivalent. However, I'm looking for a counterexample over a field which is not complete.

I have found no such counterexample neither by myself nor in the internet, but it is frequently stated that such result does not hold for non-complete fields, without a proof.

**Edit:** The stackexchange app notified me of a comment which I believe is now deleted. Still, I will address its concern in case somebody has the same idea.

I already tried to treat $\mathbb{Q}$ as a vector space over itself and find two non-equivalent norms. As for Ostrowski's theorem we know that, for example, the absolute value and the 2-adic norm are not equivalent. However, I noticed that the $p$-adic norm, though being a norm on the field $\mathbb{Q}$, it is not a norm on the vector space $(\mathbb{Q},|\cdot|)$, and thus is not a valid counterexample.