What singer Anne-Maria Hefele is performing is a style of overtone singing, and it's one of the most otherworldly sounding things I've ever heard. The whole thing is worth a watch, but the part at 3:25, where she moves the fundamental and overtone in opposite directions, had me shaking my head in disbelief.
I recommend listening to this with headphones – the overtone can seem to drop out at times, especially when played over tinny laptop speakers.
Overtone singing is also known as throat singing, of which there are several styles:
And here's what overtone singing looks like on an X-ray: