If two elements in a ED have the same Euclidean norm, are they associates? 
Is it obvious that in a Euclidean Domain two elements $x$ and $y$ having the same Euclidean norm are associates?

Can someone give me a proof of this? 

Hint $\ $ If it were true, then in a norm-euclidean quadratic field, ramification would be rampant
$$\rm\:N(w) = ww' = N(w')\:\Rightarrow\:(w) = (w')\:\Rightarrow\:(p) = (p,w)(p,w') = (p,w)^2 $$
