Holm-Kupershmidt formulation of the nondissipative part of two-fluid hydrodynamics of superfluid He-4 is supplemented with a corresponding to it dissipative part. For the complete, fully nonlinear, dynamics we prove its compatibility with thermodynamics and establish its relation to the dynamics, involving heat flux as an independent state variable, that arises in extended irreversible thermodynamics. (C) 2007 Elsevier B.V. All rights reserved.