  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.
