  Quasinormal modes are the counterparts in open systems of normal modes in
conservative systems; defined by outgoing-wave boundary conditions, they have
complex eigenvalues. The conditions are studied for a system to have a
supersymmetric(SUSY) partner with the same complex quasinormal-mode spectrum
(or the same except for one eigenvalue). The discussion naturally includes
total-transmission modes as well(incoming at one extreme and outgoing at the
other). Several types of SUSY transformations emerge, and each is illustrated
with examples, including the transformation among different Poschl-Teller
potentials and the well-known identity in spectrum between the two parity
sectors of linearized gravitational waves propagating on a Schwarzschild
background. In contrast to the case of normal modes, there may be multiple
essentially isospectral partners, each missing a different state. The SUSY
transformation preserves orthonormality under a bilinear map which is the
analog of the usual inner product for conservative systems. SUSY
transformations can lead to doubled quasinormal and total-transmission modes;
this phenomenon is analysed and illustrated. The existence or otherwise of SUSY
partners is also relevant to the question of inversion: are open wave systems
uniquely determined by their complex spectra?
