Magnetic properties of CeZn3P3, which has been believed to have the same hexagonal ScAl3C3-type crystal structure as that of a spin dimer system of YbAl3C3 at room temperature, have been investigated by magnetization, magnetostriction, specific heat (C), and magnetocaloric effect measurements. In this research, we have found that CeZn3P3 certainly has a slightly deformed crystal structure from the hexagonal one even at room temperature, which is in contrast to the structural phase transition of YbAl3C3 occuring at around 80 K, although the very slight deformation along the c plane, i.e., slightly deformed triangular lattice, and the formation of the multidomain structure are common to both compounds. CeZn3P3 is thought to maintain its deformed structure from a high-temperature region above room temperature to an extremely low-temperature region and shows a magnetic order below TN=0.8 K. The analysis of the temperature dependence of the magnetic susceptibility (χ) and C has revealed that a Kramers doublet ground state with an easy-plane type anisotropy on the c plane is well isolated from the excited states by the crystal field splitting energy of more than 300 K. Above TN, χ makes a broad peak at around 2 K, which certainly originates from the dimer formation due to the slight deformation of the triangular lattice. On the other hand, below TN, a magnetic phase diagram reminiscent of a magnetic flower blooming on the c plane was observed, which may have a close relation to a quantum effect of a quasi S=12 spin system and a contribution of the orbital component. With increasing magnetic field, we have found an anomalous magnetic state beyond the usual magnetically ordered state, where C/T is anomalously enhanced. This anomalous magnetic state is similar to that observed in YbAl3C3 induced by the field, although the magnetic ground state in YbAl3C3 is a nonmagnetic dimer state different from the normal magnetically ordered state in CeZn3P3.