Hi Ian,
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.
Lycoris squamigera 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.