Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the �cone at infinity� but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge ☆ operator twistors (i.e. vectors of the pseudo-Hermitian space H2;2) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z₂. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_p;q are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.