Your photo looks like Surprise Lily (Lycoris squamigera)
Lycoris, Nerines, Amaryllis, Hippeastrum and a number of other genera are all part of the Amaryllidaceae family. However, I am unaware of successful intergeneric crosses.
itself is thought to be a sterile triploid and probably won't breed (it is probably some freak natural hybrid, like the triploid form of Lilium lancifolium
, that propagates vegetatively quite well). You may be able to get other Lycoris
species and cross them, however (there are some known hybrids).
If you like the Lycoris you can also try Nerines. They are similar but flower a little after the Lycoris. Once again, crosses between some Nerine species should be possible.