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.

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    $\begingroup$ If you look on google you find some stuff, for example: arxiv.org/pdf/hep-th/9607086v2.pdf $\endgroup$
    – Qfwfq
    Jun 8 '13 at 15:17
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    $\begingroup$ Thank you for this reference. I also found 15 papers on arxiv containing nonassociative (or non-associative) geometry on their title, the most recent in 2013, but your reference is the first (1997). Just a note: on the 15 references found, 11 were tagged by "High Energy Physics", 2 by "Quantum algebra" and one by "Ring and algebra." It seems that the physicists are much more motivated by the nonassociative geometry than the mathematicians, why ? $\endgroup$ Jun 8 '13 at 16:18
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    $\begingroup$ Michael Atiyah speculates about the role for octonion-based geometries for quantum gravity around 30 minutes and 54 minutes into this talk: youtube.com/watch?v=I1ftUA17MZg. Also, after 58 minutes comes the thought that we need to go beyond cyclic homology for associative algebras to deal with nonassociativity. $\endgroup$ Jun 10 '13 at 10:44
  • $\begingroup$ Thank you very much David, this video is very interesting. I will put, in the issue, a quotation of Michael Atiyah extract from this video. $\endgroup$ Jun 10 '13 at 14:20

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.


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