There is progress in this direction by mathematicians here: [Jordan operator algebra][1]   

(see also this *Physics* post : [Non-associative operators in Physics](http://physics.stackexchange.com/questions/67266/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.
  


  [1]: http://en.wikipedia.org/wiki/Jordan_operator_algebra