  Supermanifolds provide a very natural ground to understand and handle
supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and
$10$ dimensions is also deeply related to the normed division algebras.
  In this paper we want to show the link between the conformal group and
certain types of symplectic transformations over division algebras. Inspired by
this observation we then propose a new\,realization of the real form of the 4
dimensional conformal and Minkowski superspaces we obtain, respectively, as a
Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a
big cell inside it.
  The beauty of this approach is that it naturally generalizes to the 6
dimensional case (and possibly also to the 10 dimensional one) thus providing
an elegant and uniform characterization of the conformal superspaces.
