  We show that normality for continued fractions expansions and normality for
base-$b$ expansions are maximally logically separate. In particular, the set of
numbers that are normal with respect to the continued fraction expansion but
not base-$b$ normal for a fixed $b\ge 2$ is
$D_2(\boldsymbol{\Pi}_3^0)$-complete. Moreover, the set of numbers that are
normal with respect to the continued fraction expansion but not normal to
\emph{any} base-$b$ expansion is $D_2(\boldsymbol{\Pi}_3^0)$-hard, confirming
the existence of uncountably many such numbers, which was previously only known
assuming the generalized Riemann hypothesis.
  By varying the method of proof we are also able to show that the set of
base-$2$ normal, base-$3$ non-normal numbers is also
$D_2(\boldsymbol{\Pi}_3^0)$-complete. We also prove an auxiliary result on the
normality properties of the continued fraction expansions of fractions with a
fixed denominator.
