You'd only run into the phrase "Gaussian integers" in a course in abstract algebra, if at all.
There's a broader definition of "algebraic integer" lurking in the background. A complex number is an algebraic integer if it is a zero of a polynomial all of whose coefficients are (ordinary) integers and whose leading coefficient is 1. Thus Sqrt[2] is an algebraic integer (x^2-2); i/2 is not (4x^2+1); and, perhaps surprisingly, (Sqrt[5]+1)/2 is (x^2-x-1).
no subject
There's a broader definition of "algebraic integer" lurking in the background. A complex number is an algebraic integer if it is a zero of a polynomial all of whose coefficients are (ordinary) integers and whose leading coefficient is 1. Thus Sqrt[2] is an algebraic integer (x^2-2); i/2 is not (4x^2+1); and, perhaps surprisingly, (Sqrt[5]+1)/2 is (x^2-x-1).