There are infinitely many ways to factor 5 in the complex numbers. Let c be a complex number. Since the complex numbers are a field, 5/c = d is a complex number. Multiply both sides by c and get 5 = cd.

However, if one restricts attention to only the Gaussian integers

https://en.wikipedia.org/wiki/Gaussian_integer, that is complex numbers like a + bi where a and b are in Z, then 5 should have a unique factorization into irreducibles (primes).

However, you seem to have two different factorizations above. I will have to check this further. Also, I am not sure why the Gaussian integers are a principal ideal domain which would make it be a unique factorization domain. So I will look up some proof for that as well.

In the case of 7, note that √6 is not an integer, that is, an element of Z. Therefore √6+i and √6-i are not Gaussian integers, but complex numbers. In the field of complex numbers 7 is a unit and everything divides it, but not in the ring of Gaussian integers.