The bright southern star L2Pup is a particularly prominent asymptotic giant branch (AGB) star, located at a distance of only 64pc. We report new adaptive optics observations of L2Pup at visible wavelengths with the SPHERE/ZIMPOL instrument of the VLT that confirm the presence of the circumstellar dust disk discovered recently. This disk is seen almost almost edge-on at an inclination of 82°. The signature of its three-dimensional structure is clearly observed in the map of the degree of linear polarization pL. We identify the inner rim of the disk through its polarimetric signature at a radius of 6AU from the AGB star. The ZIMPOL intensity images in the V and R bands also reveal a close-in secondary source at a projected separation of 2AU from the primary. Identification of the spectral type of this companion is uncertain due to the strong reddening from the disk, but its photometry suggests that it is a late K giant with comparable mass to the AGB star. We present refined physical parameters for the dust disk derived using the RADMC-3D radiative transfer code. We also interpret the pL map using a simple polarization model to infer the three-dimensional structure of the envelope. Interactions between the inner binary system and the disk apparently form spiral structures that propagate along the orthogonal axis to the disk to form streamers. Two dust plumes propagating orthogonally to the disk are also detected. They originate in the inner stellar system and are possibly related to the interaction of the wind of the two stars with the material in the disk. Based on the morphology of the envelope of L2Pup, we propose that this star is at an early stage in the formation of a bipolar planetary nebula.