  We produce new examples of totally imaginary infinite extensions of
$\mathbb{Q}$ which have undecidable first-order theory by generalizing the
methods used by Martinez-Ranero, Utreras and Videla for $\mathbb{Q}^{(2)}$. In
particular, we use parametrized families of polynomials whose roots are totally
real units to apply methods originally developed to prove the undecidability of
totally real fields. This proves the undecidability of $\mathbb{Q}^{(d)}_{ab}$
for all $d \geq 2$.
