  We use the methods of PT-symmetric quantum theory to find a one-parameter
family of ISU(1|1)-invariant planar super-Landau models with positive norm,
uncovering an `accidental', and generically spontaneously-broken, worldline
supersymmetry, with charges that have a Sugawara-type representation in terms
of the ISU(1|1) charges. In contrast to standard models of supersymmetric
quantum mechanics, it is the norms of states rather than their energies that
are parameter-dependent, and the spectrum changes discontinuously in the limit
that worldline supersymmetry is restored.
