While underwater communications have been investigated for decades, existing solutions still have difficulties in establishing reliable and low-delay links among small-size devices. The magnetic induction (MI) communication is a promising solution due to its advantages in low propagation delay and less susceptibility to surrounding environments. To date, existing MI models cannot characterize broadband and complex underwater MI channels, especially in shallow and lossy water, which significantly underestimates the underwater MI's performance. Moreover, the unpredictable polarization loss of coil antenna that makes MI unreliable has not been modeled and addressed. To this end, this paper presents a broadband channel model for underwater MI communication in complex environments. Compared with existing underwater MI models, the developed model can characterize 1) the underwater magnetic field propagation at any point in the 3-D space between the water surface and water floor, 2) both the near and far fields of all feasible underwater signal bands, 3) the impacts of lossy underwater medium on not only the propagation path but on the MI antenna itself as well, and 4) the arbitrarily orientated tri-directional coil antenna that can eliminate the MI antenna's susceptibility to orientation changes. The developed model is validated through COMSOL-multiphysics simulations and in-lab experiments.