  Let $A$ be an abelian variety over a number field. The connected monodromy
field of $A$ is the minimal field over which the images of all the $\ell$-adic
torsion representations have connected Zariski closure. We show that for all
even $g \geq 4$, there exist infinitely many geometrically nonisogenous abelian
varieties $A$ over $\mathbb{Q}$ of dimension $g$ where the connected monodromy
field is strictly larger than the field of definition of the endomorphisms of
$A$. Our construction arises from explicit families of hyperelliptic Jacobians
with definite quaternionic multiplication.
