Every Hilbert space H has an orthonormal basis. This basis is a Hamel basis if and only if H is finite-dimensional. The algebra of bounded linear operators on H, B(H), is quite complicated, but its quotient over the ideal of compact operators (the Calkin algebra) is even worse (or better, depending on one’s taste) than P(N)/Fin. All of these statements fail in some models of ZF in which the Axiom of Choice fails. This is a report on joint work in progress with Bruce Blackadar and Asaf Karagila.