It follows that u+vi is a factor of x+yi iff u-vi is a factor of x-yi. If you have the complete factorization for a complex number z, conjugate all those primes to get the unique factorization for the conjugate of z.
If a gaussian prime p happens to be a pure integer, and it divides x+yi, the quotient has to be (x/p)+(y/p)i, hence p divides both x and y.
If u+vi is a complex prime, could its conjugate, u-vi, be the same prime? If they are associates of each other, u+vi is multiplied by 1, -1, i, or -i, to get u-vi. If 1 or -1 is used, that forces u or v to zero. But u+vi is a complex prime, so both coordinates are nonzero. Using i or -i forces u = ±v. Now u±ui can only be prime if u = 1. In fact 1+i is prime, and its conjugate, 1-i, is the same prime, merely 1+i × -i. This is the only prime with 4 associates. All other primes have two associates, the prime and its opposite. This could be p and -p, or u+vi and -u-vi.
If the complex prime u+vi divides the real number n, take conjugates to show u-vi also divides n. If u = v = 1 then (1+i)×(1-i), or 2, divides n, and n is even. For any other values of u and v, u+vi and u-vi are different primes. They both divide n, hence n is divisible by u2+v2.