The optical emission of cone-shell quantum structures (CSQS) under vertical electric (F) and magnetic (B) fields is studied by means of simulations. A CSQS has a unique shape, where an electric field induces the transformation of the hole probability density from a disk into a quantum-ring with a tunable radius. The present study addresses the influence of an additional magnetic field. A common description for the influence of aB-field on charge carriers confined in a quantum dot is the Fock-Darwin model, which introduces the angular momentum quantum numberlto describe the splitting of the energy levels. For a CSQS with the hole in the quantum ring state, the present simulations demonstrate aB-dependence of the hole energy which substantially deviates from the prediction of the Fock-Darwin model. In particular, the energy of exited states with a holelh>0 can become lower than the ground state energy withlh=0. Because for the lowest-energy state the electronleis always zero, states withlh>0 are optically dark due to selection rules. This allows switching from a bright state (lh=0) to a dark state (lh>0) or vice versa by changing the strength of theForBfield. This effect can be very interesting for trapping photoexcited charge carriers for a desired time. Furthermore, the influence of the CSQS shape on the fields required for the bright to dark state transition is investigated.