The non-associativity of octonions is the basic problem if one attempts to build octonionic quantum mechanics. Nothing like this is tried in TGD. Instead, classical number fields appear at the level of classical physics (see this). Space-time surfaces as classical correlates of quantum physics are conjectured to decompose to associative (/quaternionic/Minkowskia)n and co-associative (/co-quaternionic/Euclidian) regions so that the weakness of octonionic quantum mechanics would turn into a strength making classical physics completely unique purely number theoretical. More precisely, the induced spinor structure for 8-D imbedding space has a special representation in terms of octonionic gamma matrices and the induced gamma matrices (not strictly speaking matrices anymore) are conjectured to span a quaternionic or co-quaternionic subspace of octonions over complex numbers at each point of the preferred extremal of Kähler action.
Addition: Also Motl has comments about octonions. The usual flood of insults and extremely arrogant super-stringy attitude towards anyone who does not regard superstrings as the laws of Moses for physics and dares to ask whether some aspects of super-strings might be part of a more successful physical theory. John Baez was the target of the aggression at this time. Maybe it is high time to Lubos to realize that the glamour of Harward does not last forever: we also remember that the exit of Lubos from Harward was not graceful. Some real output would be desperately needed if Lubos wants to keep his position as a blog authority and we have been waiting for years. My comment about the role of classical number fields in physics of course goes un-noticed: Lubos reads nothing which he has decided to represent crackpot theory. In any case, Lubos does a valuable work: he teaches us to tolerate people behaving like complete idiots. Learning this is after all the only manner to build a better world;-).