Equivalence of Gaussian prime numbers

Consider two Gaussian integers a and b. a=x+iy and b=c+id in the Cartesian plane.
Any C=ab then C is the linear combination of (x+iy)and (−y+ix) similarly it is also a linear combination of (c+id) and (−d+ic).
In general one can tell that C falls on the grid created by vector 'a' and 'ia' in Cartesian plane.
Any Gaussian integer has a number of residues equal to its norm; it can be easily proved by the graphical method.
In general any Gaussian integer has number of residues equal to its norm. The Gaussian primes having their norms as a prime number can be assigned to those prime numbers (norm).
Caley tables of that Gaussian prime number and its norms are equivalent.
hint: prove it also by graphical method.
In the sense it is a direct proof that many one dimensional properties are valid for such a Gaussian prime,isomorphism.
fermats little theorem in complex plane :
let p be a 1D prime of the form (4n+1),then in 2D (in gaussian system) any complex number(x+iy)^ p is congruent to (x+iy) modulo p, as in 1D.
but if p is a prime of the form 4n+3, then in 2D (in gaussian system) any complex number(x+iy)^ p is congruent to (x-iy) modulo p, hint:prove it by binomial theorem.

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