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.
