I can think of all kinds of non-commutative operations that have real world significance. Rotations are an easy example, all you have to do is spend a minute with a Rubik's cube and you'll see how rotation order matters. If you think rotation isn't combined with multiplication, remember complex numbers do just that, before commutativity is even a factor (no pun intended.)

In case you're having trouble with i and j being equal to the same sqrt(-1), here's a tip: define u^2=a and you can see that the inverse already has two values without leaving the reals. If you defined u=sqrt(a), that wouldn't be true. (If you explore the other solutions you'll eventually get to octonions, which don't even have associativity.)