The Hurwitz-Frobenious theorem is your friend here. It states that the only normed division algebras over the real numbers are the reals, the complex numbers which are obtained by doubling the reals, the quaternions that are obtained by doubling the complex numbers, and the octonions that are obtained by doubling the quaternions.
With each doubling, you loose something. You loose ordering moving to the complex numbers. You loose commutativity moving to the quaternions, and you loose associativity moving to the octonions.
You can continue the process to get the Clifford algebras with dimension 2^n, but you loose the existence of inverses for all non-zero elements, and so these are no longer normed division algebras.
The doubling process is mechanically isomorphic to constructing 2x2 matrices from two elements of the previous algebra with with the diagonal duplicating one value 'a' and the cross diagonal conjugating the other value 'b'.
Geometrically, the quaternions are a four dimensional vector space with Euclidean norm that admits a well defined multiplication with inverses. This multiplication is a representation of the four dimensional rotations (which rotate a plane through the origin and fix the two dimensions orthogonal to that plane). The three dimensional rotations are a proper subset of the four dimensional rotations.
Any plane through the origin within the quaternions is algebraically isomorphic to the complex numbers, which in turn may be thought of as a representation of the two dimensional rotations.
These are the basic facts, consideration of which should serve to clarify your terminology and answer your questions.