A twistor-like extension of the $N=2$ conformal superspace (SSp) with nontrivial central charges is presented. Parameters of new (anti)commutative generators carrying an index of the spin $SU(2)$-symmetry are used to construct massless harmonics, dramatically reducing the number of essential coordinates. Under this approach, the usual vector coordinates become complex, while the Grassmann coordinates are reduced to a single variable which is scalar under the Lorentz transformation. The choice of the appropriate analytical subspace in the initial SSp is rather difficult, but our closed algebra gives the possibility of covariantly reducing the theory from the initial SSp to an SSp of a smaller Grassmann dimension. A special definition of the coordinates and the flat space for this structure allows one to obtain the Penrose twistor equation and the Ogievetsky harmonics.