The twisted Kloosterman sums over Z were studied by V. Bykovsky, A.Vinogradov, N. Kuznetsov, R. W. Bruggeman, R. J. Miatello, I. Pacharoni, A. Knightly, and C. Li. In our paper, we obtain similar estimates for Kχ(α, β; γ; q) over ℤ[i] and improve the estimates obtained for the sums of this kind with Dirichlet character χ (mod q1), where q1 | q.
