How about basketball shoes for badminton?

Depent on the level of games & the hall.
Here on fun casual games you could see someone wearing running shoes.
On some less strict place you could even see someone wearing sandal or even barefoot. They are fine as their level of games are pretty much beginner & fun stroke only.

But on serious games, most likely they wear badminton shoes tho the pricing vary alot. Some wear a cheap one, some an expensive one.

Regarding basketball shoes. The trait of basketball are jumping. So it had better ankle protection tho some of it are somewhat to much for badminton flexible move. In term of cushion it provide good absorbtion as its design to nullify impact from high jump but its heel tend to be higher than badminton shoes thus makes it less stable.