Yes, you can play badminton with normal shoes, but you will not be able to move as quickly or jump as high as you would if you were wearing special badminton shoes.  Normal shoes, especially those with rough soles, could cause injury to both yourself and your teammates. The risk of injury is especially high during scrums, when the ball bounces unexpectedly off the court surface. That being said, it is possible to play badminton with normal shoes if they have the required features mentioned above.