What are some examples of a non commutative division ring other than quaternions?
Noncommutative domains which satisfy the right Ore condition allow you to build a "right division ring of fractions" in an analogous way to that of the field of fractions for a commutative domain.
This division ring is necessarily not commutative if you pick the domain to be not commutative :) Not commutative right Ore domains are pretty easy to come by: in particular, right Noetherian domains are right Ore. So for example, you could look at the division ring of quotients for $\Bbb H[x]$.
You can find many examples in:
R. S. Pierce, Associative algebras,Springer-Verlag, 1982