I think if you want to study quantum computing, you do not need as much of a physics background as you think (perhaps for physically implementing a qc, but that is more experimental).
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).
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).