© 2015, Pleiades Publishing, Ltd. Spherical waves in which one of the coordinates of a source point is complexified are considered. Interest in such exact solutions of the wave equation known as “complex source wave fields,” is stipulated by their Gaussian localization, both in the time-harmonic regime and in the nonstationary case, under a proper choice of the wave form. Since a correct description of the square root occurring in the solution requires the choice of a branch cut, the solution has a jump in the physical space and thus satisfies an equation with a certain source. We study such sources in the physical space for various choices of a branch of the root for the time-harmonic and nonstationary cases. From this point of view, Izmest’ev—Deschamps Gaussian beams, Gaussian wave packets, solutions presented by Felsen and Heyman, Sheppard and Saghafi, Saari, as well as X-waves, are examined.