The author introduces and studies two analogues of the general Kloosterman sum (containing multiplicative characters) over the ring \(\mathbb Z[i]\). Several estimates of the sums are obtained. As the author states, one of the sums (which has no counterparts among trigonometric sums over \(\mathbb Z\)) can be used for the investigation of the second moment of the Hecke zeta function of the field \(\mathbb Q(i)\).