We consider the complete Euler system describing the time evolution of a general inviscid compressible fluid. In view of the latest non-uniqueness results for admissible weak solutions we introduce a new concept of measure-valued solution based on the total energy balance and entropy inequality for the physical entropy. We study the system in conservative variables usual for numerical analysis. Our class of so-called dissipative measure-valued solutions satisfies the weak-strong uniqueness property and it is large enough to include the vanishing dissipation limits of the Navier-Stokes-Fourier system. Furthermore, we introduce the concept of maximal dissipative measure-valued solution to our system. These are solutions that maximize the entropy production rate. We show that these solutions exist under fairly general hypotheses imposed on the data and constitutive relations.