It is a non-abelian extension of the complex field - multiplication is not commutative. They can be described in terms of pauli spin matrices which is partly why physicist start to dribble unbecomingly when they come up in conversation.
They are the set of all things of the form a+bi+cj+dk where a,b,c,d are real numbers and i,j,k satisfy ij = k jk=i ki=j i^2=j^2=k^2=-1=ijk
or they are they can be realized as a degree two extension of the complx numbers.
C is algebraically closed is exactly what you should have thunk.
the quaternions aren't a filed, they are a division algebra.