We investigate the polaron and molecular states of a fermionic atom with one-dimensional spin-orbit coupling(SOC)coupled to a three-dimensional spinless Fermi sea.Because of the interplay among the SOC,Raman coupling and spin-selected interatomic interactions,the polaron state induced by the spin-orbit coupled impurity exhibits quite unique fea-tures.We find that the energy dispersion of the polaron generally has a double-minimum structure,which results in a finite center-of-mass(c.m.)momentum in the ground state,different from the zero-momentum polarons where SOC are introduced into the majority atoms.By further tuning the parameters such as the atomic interaction strength,a discon-tinuous transition between the polarons with different c.m.momenta may occur,signaled by the singular behavior of the quasiparticle residue and effective mass of the polaron.Meanwhile,the molecular state as well as the polaron-to-molecule transition is also strongly affected by the Raman coupling and the effective Zeeman field,which are introduced by the lasers generating SOC on the impurity atom.We also discuss the effects of a more general spin-dependent interaction and mass ratio.These results would be beneficial for the study of impurity physics brought by SOC.