This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math.
The associative property is astonishingly fundamental. More than once I've heard musings that it isn't clear if the associative property is fundamental to the universe, or merely fundamental to how the brain works. While non associative algebgras certainly exist, it is the associative ones that seem to form the backbone of mathematics.
When studying non associative algebras, it is common to first embed the algebra in an associative one first and then do the reasoning there. It is my belief (please correct me if I'm wrong) that the syntactic reasoning of proof theory is constructed using associative data structures and operations (strings with associative operations like concatenation).
Do we ever study a non associative algebra without somehow framing it inside an associative approach? I ask because I have not been able to imagine what that might look like. If we do, what do we gain by such an approach?