A set {a,b,c,d} of four non-zero distinct polynomials in
Z[i][X] is said to be a Diophantine D(4)-quadruple if the product
of any two of its distinct elements increased by 4 is a square of some
polynomial in Z[i][X].
In this paper we prove that every D(4)-quadruple in Z[i][X] is
regular, or equivalently that the equation (a+b−c−d)2=(ab+4)(cd+4) holds
for every D(4)-quadruple in Z[i][X].Comment: some parts were reorganized, corrections mad