We study the homotopy types of certain spaces closely related to the spaces of algebraic (rational) maps from the m dimensional real projective space into the n dimensional complex projective space for 2 ≤ m ≤ 2n (we conjecture this relation to be a homotopy equivalence).In [9] we proved that natural maps of these spaces to the spaces of all continuous maps are Z/2-equivariant homotopy equivalences, where the Z/2-equivariant action is induced from the conjugation on C. In the same article we also proved that the homotopy types of the terms of the natural degree filtration approximate closer and closer the homotopy type of the space of continuous maps and obtained bounds that describe the closeness of the approximation in terms of the degrees of the maps. In this paper, we improve the estimates of the bounds by using new methods invented in [13] and used in [10].In addition, in the the last section, we prove a special case (m = 1) of the conjecture stated in [1] that our spaces are homotopy equivalent to the spaces of algebraic maps.