The Drude model for metal is extended to include complex relaxation rates. As a test for what happens to the surface plasmon resonances with such metals, the lifetime is examined for propagating waves across a single planar metal-dielectric interface. By analytically solving the dispersion relation being fourth-order in the complex frequency, group-velocity dispersion and quality factors are explicitly found. Due to the symmetry breaking between the forward and backward waves, standing waves are not allowed in general.