Return to Answer

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.

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.

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.

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.

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

(see also this Physics post : Non-associative operators in Physics)

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.