  Supersymmetric quantum mechanics is constructed in a new non-Hermitian
representation. Firstly, the map between the partner operators $H^{(\pm)}$ is
chosen antilinear. Secondly, both these components of a super-Hamiltonian
${\cal H}$ are defined along certain topologically nontrivial complex curves
$r^{(\pm)}(x)$ which spread over several Riemann sheets of the wave function.
The non-uniqueness of our choice of the map ${\cal T}$ between "tobogganic"
partner curves $r^{(+)}(x)$ and $r^{(-)}(x)$ is emphasized.
