  We present new constructions for perfect and odd perfect sequences over the
quaternion group $Q_8$. In particular, we show for the first time that perfect
and odd perfect quaternion sequences exist in all lengths $2^t$ for $t\geq0$.
In doing so we disprove the quaternionic form of Mow's conjecture that the
longest perfect $Q_8$-sequence that can be constructed from an orthogonal array
construction is of length 64. Furthermore, we use a connection to combinatorial
design theory to prove the existence of a new infinite class of Williamson
sequences, showing that Williamson sequences of length $2^t n$ exist for all
$t\geq0$ when Williamson sequences of odd length $n$ exist. Our constructions
explain the abundance of Williamson sequences in lengths that are multiples of
a large power of two.
