Algebraic formulas are algebraic formulas...
Substituting Gaussian integers z_{1} and z_{2} into the usual parametric formulas for Pythagorean triples,
(A, B, C) = (z_{1}^{2}  z_{2}^{2}, 2*z_{1}*z_{2}, z_{1}^{2} + z_{2}^{2})
We assume that z_{1} and z_{2} are nonzero. We obtain primitive triples if gcd(z_{1}, z_{2}) = 1 and gcd(z_{1} + z_{2}, 2) = 1. The latter condition rules out z_{1} and z_{2} being complexconjugate.
We obviously obtain thinly disguised versions of rationalinteger triples when one of z_{1} and z_{2} is real, and the other is pure imaginary.
Obviously A, B, and C are real when z_{1} and z_{2} are rational integers.
Clearly B is real when z_{2} is a real multiple of conj(z_{1}).
Also, B/C is real when z_{2}/z_{1} is real, or z_{1} = z_{2}.
A/C is only real when z_{2}/z_{1} is real.
The nontrivial primitive solutions with A, B, C all complex having the smallest coefficients appear to be
z_{1} = 1, z_{2} = 1 + I: A = 1  2*I, B = 2 + 2*I, C = 1 + 2*I
and variants.
