  Given an infinite word over the alphabet $\{0,1,2,3\}$, we define a class of
bipartite hereditary graphs $\mathcal{G}^\alpha$, and show that
$\mathcal{G}^\alpha$ has unbounded clique-width unless $\alpha$ contains at
most finitely many non-zero letters.
  We also show that $\mathcal{G}^\alpha$ is minimal of unbounded clique-width
if and only if $\alpha$ belongs to a precisely defined collection of words
$\Gamma$. The set $\Gamma$ includes all almost periodic words containing at
least one non-zero letter, which both enables us to exhibit uncountably many
pairwise distinct minimal classes of unbounded clique width, and also proves
one direction of a conjecture due to Collins, Foniok, Korpelainen, Lozin and
Zamaraev. Finally, we show that the other direction of the conjecture is false,
since $\Gamma$ also contains words that are \emph{not} almost periodic.
