GaussianIntegers is an option for FactorInteger, PrimeQ, Factor and related functions which specifies whether factorization should be done over Gaussian integers.
With GaussianIntegers -> False, factorization is done over the ordinary ring of integers .
With GaussianIntegers -> True, factorization is done over the ring of integers with adjoined .
Example: FactorInteger[13, GaussianIntegers -> True] .
The Gaussian primes used when GaussianIntegers -> True are chosen to have both real and imaginary parts positive.
The first entry in the list given by FactorInteger with GaussianIntegers -> True may be -1 or -I.
See The Mathematica Book: Section 3.2.4.
See also: Extension, ComplexExpand.