The structure of quasi-2d solutions of dipolar superballs in the fluid state has been determined by Metropolis Monte Carlos simulations in the absence and the presence of an external field. Superballs are 3d objects characterized by a one shape parameter. Here, superballs resembling cubes, but possessing rounded edges, have been used. Examination has been made for several magnitudes of the dipole moment in three different dipole directions. In the limit of a cube, the directions become (i) the center of mass - the center of a face (001) direction, (ii) the center of mass - the center of an edge (011) direction, and (iii) the center of mass - the corner (111) direction. At a small dipole moment, the superballs are translationally and orientationally disordered, and the dipoles become partially orientationally ordered in the presence of the field parallel to the plane of the superballs. At a large dipole moment, chains of superballs are formed, and the chains become parallel in the presence of the field. The chains remain separated for the dipole in the 001-direction and form bundles for the 011- and 111-directions. The different structures obtained for the different dipole directions are interpreted in terms of how compatible the dipole-dipole interaction is with the cube-cube interaction at short separation for the different directions of the dipole moment. Hence, the structural richness arises from an interplay of the different symmetries of a cube and of the field of a dipole.