  We elaborate the definition and properties of ''massive" elementary systems
in the (1+3)-dimensional Anti-de Sitter (AdS$_4$) spacetime, on both classical
and quantum levels. We fully exploit the symmetry group Sp$(4,\mathbb R)$, that
is, the two-fold covering of SO$_0(2,3)$ (Sp$(4,\mathbb R) \sim$
SO$_0(2,3)\times \mathbb Z_2$), recognized as the relativity/kinematical group
of motions in AdS$_4$ spacetime. In particular, we discuss that the group coset
Sp$(4,\mathbb R)$/S(U(1)x SU(2)), as one of the Cartan classical domains, can
be interpreted as a phase space for the set of free motions of a test massive
particle on AdS$_4$ spacetime; technically, in order to facilitate the
computations, the whole process is carried out in terms of complex quaternions.
The (projective) unitary irreducible representations (UIRs) of the
Sp$(4,\mathbb R)$ group, describing the quantum version of such motions, are
found in the discrete series of the Sp$(4,\mathbb R)$ UIRs. We also describe
the null-curvature (Poincar\'{e}) and non-relativistic (Newton-Hooke)
contraction limits of such systems, on both classical and quantum levels. On
this basis, we unveil the dual nature of ''massive" elementary systems living
in AdS$_4$ spacetime, as each being a combination of a Minkowskian-like massive
elementary system with an isotropic harmonic oscillator arising from the
AdS$_4$ curvature and viewed as a Newton-Hooke elementary system. This
matter-vibration duality will take its whole importance in the quantum regime
(in the context of the validity of the equipartition theorem) in view of its
possible r\^{o}le in the explanation of the current existence of dark matter.
