We study the supersymmetric Casimir energy E susy of N = 1 backslashmathcalN=1 field theories with an R-symmetry, defined on rigid supersymmetric backgrounds S 1 texttimesM 3, using a Hamiltonian formalism. These backgrounds admit an ambi-Hermitian geometry, and we show that the net contributions to E susy arise from certain twisted holomorphic modes on ℝthinspacetexttimes M 3, with respect to both complex structures. The supersymmetric Casimir energy may then be identified as a limit of an index-character that counts these modes. In particular this explains a recent observation relating E susy on S 1 texttimes S 3 to the anomaly polynomial. As further applications we compute E susy for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.