A All complex integers of the same norm = associates?

Are all complex integers that have the same norm associates of each other?

I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a satisfying definition of associate as part of my research on the Gaussian and Eisenstein integers.

I have found a satisfying definition of a Gaussian prime or an Eisenstein prime as being "only divisible by a root of unity and one of its associates". But now I want to find a definition of "associate". The definition I found is "a multiple of the number with one of the roots of unity". This is a circular definition. If the definition of the associate of a Gaussian or Eisenstein integer p were "all integers that have the same norm as p", then that would be great. But since I haven't found this definition, I wonder if I am missing something?

Thanks!
-Jeffrey
 
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I have found a satisfying definition of a Gaussian prime or an Eisenstein prime as being "only divisible by a root of unity and one of its associates". But now I want to find a definition of "associate". The definition I found is "a multiple of the number with one of the roots of unity". This is a circular definition.
It is not circular. In the Gaussian integers the roots of unity are 1, i, -1 and -i. The associates of a number z are z, iz, -z, -iz. Now you can use this definition to define primes.

Using your definition, 3+4i, 3-4i and 5 are not associates, despite having the same norm.
 

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