Are there any Euclidean Rings with a norm which is not additive nor multiplicative?

Question

We note that a Euclidean Domain is an integral domain $R$ together with a norm function $N:R\setminus\{0\}\rightarrow \mathbb{Z}_{\geq0}$ such that for $a,b\in R$ with $b\neq 0$, there exists $q,r\in R$ such that $a=bq+r$ where $r=0$ or $N(r)<N(b)$. Furthermore, let us call the norm multiplicative if $N(ab)=N(a)N(b)$ and additive if $N(ab)=N(a)+N(b)$. I feel like most examples of Euclidean rings that I know of have either a multiplicative or additive norm. For example, $\mathbb{Z}[i]$ has a multiplicative norm, or $K[x]$ where $K$ is a field has an additive norm. However, I don't think I know of any examples of Euclidean Rings which have a norm that is not additive nor multiplicative. I was curious if there are any easy or natural examples, or if every (one that we care about) has a multiplicative or additive norm.