Both formalisms of a chiral multiplet in Supersymmetry

For the description of a chiral multiplet in Supersymmetry there are 2 formalisms, the one I am used to presented for instance in the Supersymmetry Primer of SP Martin which is based on 2-component (Weyl)-spinors and another one which apparently tries to avoid the intricateness of the Weyl-spinor formalism.

For instance in the book on Supergravity from Freedman and van Proeyen it is written on the 4-spinor based formalism (page 109):

… the most basic SUSY field theories are:
the chiral multiplet, which contains a self-conjugate spin-1/2 fermion described by the Majorana field $chi(x)$ plus a complex spin-$0$ boson described by a scalar field $Z(x)$. Alternatively $chi(x)$ may be replaced by the Weyl spinor $P_Lchi$ and/or $Z(x)$ by the combination $Z(x) =frac{A(x) + i B(x)}{sqrt{2}}$
where $A$ and $B$ are a real scalar and a pseudo-scalar, respectively.

$P_L$ is a projection operator that projects out the left-chiral part of the spinor field.

How can be $Z(x)$ be a combination of a scalar and a pseudo-scalar field?
And why the real part is "scalar" and the imaginary part "pseudo-scalar"?
Upon the introduction of the chiral multiplet by Martin’s Supersymmetry Primer no word is said on the symmetry properties of the complex scalar field and how it could be decomposed. Moreover, the 4-spinor formalism of the chiral multiplets seems to suggest a double number of degrees of freedom (dof) for the fermion whereas the number of dof’s of the scalar field seems to be unchanged, but kind of suggesting that in the Weyl-formalism one would work with the fields $A$ and $B$ separately which is apparently not the case, but rather confusing.

So my main question is: How can $Z(x)$ be a combination of scalar and pseudo-scalar field (and how they are distributed on the real and imaginary part).
Any help is greatly appreciated.