I'm not sure if this is exactly what you are looking for or perhaps you already know what I am about to say.

There is a geometric notion of a twistor spinor (or conformal Killing spinor): one which is in the kernel of the Penrose operator (see below). Then one defines the twistor space as the projectivisation of the space of twistor spinors. Doing this for Minkowski spacetime recovers the usual twistor space.

Let $(M,g)$ be a riemannian spin manifold. (When I say riemannian I include also the case of a metric with indefinite signature.) Let $S$ denote the complex spinor bundle. The spin connection defines a map
$$
\nabla: \Gamma(S) \to \Omega^1(S)
$$
from spinor fields to one-forms with values in $S$. Now $\Omega^1(S) = \Gamma(T^*M \otimes S)$ and Clifford action of one-forms on spinors gives a map
$$
\Omega^1(S) \to \Gamma(S)
$$
The composition of the previous two maps is the Dirac operator. The Penrose operator is in some sense the complement of the Dirac operator $D$. The kernel of the Clifford map $T^*M \otimes S \to S$ defines a subbundle $W$, say, of $T^*M \otimes S$. Composing the covariant derivative with the projection $\Omega^1(S) \to \Gamma(W)$ defines the Penrose operator $P: \Gamma(S) \to \Gamma(W)$: explicitly,
$$
P_X \psi = \nabla_X \psi + \frac1n X \cdot D\psi
$$
for all vector fields $X$ and spinor fields $\psi$, and where $n = \dim M$. (My Clifford algebra conventions are $X^2 = - |X|^2$.) Notice that the "gamma trace" of the Penrose operator vanishes.

There is a sizeable literature on twistor spinors mostly in riemannian and lorentzian signatures. This is the work of Helga Baum and collaborators in Berlin. A search for "twistor spinors" in MathSciNet should give you many links.

One important property of the twistor spinor equation is that it is conformally invariant, whence the twistor spinors of conformally related riemannian spin manifolds correspond in a simple way. Since you mention maximally symmetric lorentzian manifolds, this observation might be of use because such spaces are conformally flat, hence you can write down the twistor spinors simply by rescaling the twistor spinors in Minkowski spacetime. In riemannian signature (hence for round spheres and hyperbolic spaces) this is described in the 1990 Humboldt University Seminarberichte *Twistor and Killing spinors on riemannian manifolds* by Baum, Friedrich, Grunewald and Kath, later published by Teubner.

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