The generalization of an elliptical Gaussian beam is presented by introducing a complex shift in the transverse coordinates. The intensity distribution of the resulting beam is a Gaussian function distribution with the width varying as an ordinary elliptical Gaussian beam in the plane perpendicular to z axis, the peak intensity of which traces a linear trajactory. The wave fronts are displaced laterally. The beam is referred to as a decentered elliptical Gaussian beam, its transformation is deduced quantitatively when passing through an optical system.