There has been progress in this area by mathematicians, as seen here: Jordan Operator Algebra.
(See also this Physics post: Non-associative Operators in Physics).
Note: Jordan operator algebras replace associativity with commutativity. Specifically, their product $\circ$, defined as $a_1 \circ a_2 = \frac{1}{2}(a_1*a_2 + a_2*a_1)$, is commutative but nonassociative, while $*$ is noncommutative and associative. This approach is advanced, yet not entirely satisfying. A truly comprehensive solution would involve algebras that are both nonassociative and noncommutative.