"We introduce an alternative factorization of the Hamiltonian of the quantum harmonic oscillator which leads to a two-parameter self-adjoint operator from which the standard harmonic oscillator, the one-parameter oscillators introduced by Mielnik, and the Hermite operator are obtained in certain limits of the parameters. In addition, a single Bernoulli-type parameter factorization which is different of the one introduced by M. A. Reyes, H. C. Rosu, and M. R. Gutie´rrez, Phys. Lett. A 375 (2011) 2145 is briefly discussed in the final part of this work."