At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below:
Audience: You showed that noncommutative geometry greatly simplifies the physics in a very elegant way, by actually losing a property that seems a bit simple. Similary, if we work with algebras, complex numbers, quaternions and octonions..., are there people investigating some nonassociative geometries? Alain Connes: Ok, I can explain you about this right now...
The video cuts off at that time, even before Alain finishes his explanation, so I post this question here :
What is about nonassociative geometry ?
Sir Michael Atiyah (here at 58'): Connes' theory is very beautiful, but it only deals with associative algebras, so it can't deals with the octonions, so that's why I think this theory is not quite finished.