A discussion on the q-oscillator system and the algebra Hq(4) is presented, emphasizing the case of |q| = 1 in particular. A remarkable connection between algebras SUq(2) and Hq(4) is given. The single q-oscillator system is solved and shown to be analogous to the parafermion-(PF), with the specific cases analogous to 2nd and 3rd order PF's discussed in detail. The properties of the representations of Hq(4) and a finite chain model of q-oscillators are discussed-