Abstract: Elastic waves are affected by viscoelasticity during the propagation through the Earth, resulting in energy attenuation and phase distortion, in turn resulting in low seismic imaging accuracy. Therefore, viscoelasticity should be considered in seismic migration imaging. We propose a Q compensated multi-component elastic Gaussian beam migration (Q-EGBM) method to (1) separate the elastic-wave data into longitudinal (P) and transverse (S) waves to perform PP-wave and PS-wave imaging; (2) recover the amplitude loss caused by attenuation; (3) correct phase distortions caused by dispersion; (4) improve the resolution of migration imaging. In this paper, to accomplish (2), (3), and (4), we derive complex-valued traveltimes in viscoelastic media. The results of numerical experiments using a simple five-layer model and a sophisticated BP gas model show that the method presented here has significant advantages in recovering energy decay and correcting phase distortion, as well as significantly improving imaging resolution.