  A set $\{a, b, c, d\}$ of four non-zero distinct polynomials in
$\mathbb{Z}[i][X]$ 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 $\mathbb{Z}[i][X]$.
  In this paper we prove that every $D(4)$-quadruple in $\mathbb{Z}[i][X]$ is
regular, or equivalently that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds
for every $D(4)$-quadruple in $\mathbb{Z}[i][X]$.
