In memory of Marcos Moshinsky, who promoted the algebraic study of the harmonic oscillator, some results recently obtained on an infinite family of deformations of such a system are reviewed. This set, which was introduced by Tremblay, Turbiner, and Winternitz, consists in some Hamiltonians H k on the plane, depending on a positive real parameter k. Two algebraic extensions of H k are described. The first one, based on the elements of the dihedral group D 2k and a Dunkl operator formalism, provides a convenient tool to prove the superintegrability of H k for odd integer k. The second one, employing two pairs of fermionic operators, leads to a supersymmetric extension of H k of the same kind as the familiar Freedman and Mende super-Calogero model. Some connection between both extensions is also outlined.