Generalizing the finiteness problem for non-rational starting points

Determine the correct generalization of the finiteness question for parameters in the quadratic family f_c(x)=x^2+c when the starting point a is algebraic but not rational. Specifically, ascertain the appropriate notion of "totally real" for parameters c (with respect to Q versus Q(a)) so that the problem of deciding finiteness of the set {c in the chosen totally real field: f_c^n(a)=f_c^m(a) for some integers 0 ≤ m < n} is well-posed for a in the algebraic closure of Q minus Q.

Background

The paper proves that for rational a with |a|<2 the set of totally real algebraic parameters c such that a is preperiodic under f_c(x)=x^2+c is finite, and for |a|≥2 it is infinite. It then raises a question about extending this finiteness classification beyond rational a.

The authors note a subtlety: when a is not rational, the notion of “totally real” for c can be taken relative to Q or relative to Q(a), and these can differ. They give the example a=√2−1 and c=−2+√2, where c is preperiodic for a but the Galois conjugate of c over Q is not, illustrating that the base field matters for conjugation and the “totally real” property.