Only in the sense that they are both require four real coefficients. The quaternions have a particular multiplicative structure that just doesn't apply to quantum states, so it doesn't make sense to use them.

That being said, the space of single-qubit operations is very much analogous to rotations in 3d and so is well described by quaternions. In fact, the Pauli matrices times i (iX,iY,iZ) are isomorphic to the quaternions (i,j,k). For example, iX * iY * iZ = -I.

If you have had more exposure to maths and computer science, it will be easier for you than someone with a "pure" physics background.

As for quarternions, yes they are isomorphic, but generally for useful applications, people consider quantum computers with n qubits. So your state is an element of C^(2^n). Apart from the measurement step, you can idealise any quantum computation as a unitary transformation, so an element of the unitary group U(2^n), acting on this complex vector.

An element of U(2^n) is representable as a 2^n x 2^n matrix U, with complex entries, st U.U^{\dagger} = I. Here dagger represents conjugate transpose, and I is the 2^n x 2^n identity matrix. Sometimes people add the extra constraint, det(U) = 1, then this gives you the special unitary group SU(2^n).