  In this paper we investigate a particular possibility to extend C(1,3)
conformal symmetry using Heisenberg operators, and a related possibility to
extend conformal supersymmetry using parabose operators. The symmetry proposed
is of a simple mathematical form, as is the form of necessary symmetry breaking
that reduces it to the conformal (super)symmetry. It turns out that this
extension of conformal superalgebra can be obtained from standard non-extended
conformal superalgebra by allowing anticommutators $\{Q_\eta, Q_\xi\}$ and
$\{\bar Q_{\dot \eta}, \bar Q_{\dot \xi}\}$ to be nonzero operators and then by
closing the algebra. In regard of the famous Coleman and Mandula theorem (and
related Haag-Lopuszanski-Sohnius theorem), the higher symmetries that we
consider do not satisfy the requirement for finite number of particles with
masses below any given constant. However, we argue that in the context of
theories with broken symmetries, this constraint may be unnecessarily strong.
