uniqueness Euclidean Norm in Euclidean domain

Question

Let $R$ be a Euclidean domain with the euclidean norm $N$. Is $N$ unique?

Is there any trivial counterexample? or is that statement is somehow true?

Answer 1

I Found a counter example:

For the ring of polynomials, we can take $N_1(p)=\deg(p)$ and $N_2(p)=2^{\deg(p)}$.