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.
Best Answer
There is progress in this direction by mathematicians here: Jordan operator algebra
(see also this Physics post : Non-associative operators in Physics)
Warning: the Jordan operator algebras exchange the associativity by the commutativity, in fact their product $\circ$, given by $a_1 \circ a_2 = \frac{1}{2}(a_1*a_2+a_2*a_1)$, is commutative nonassociative, whereas $*$ is noncommutative associative. So it's an advanced but it's not really satisfying.
A satisfying advanced would be with nonassociative and noncommutative algebras.