We show that there are Turing complete computably enumerable sets of
arbitrarily low non-trivial initial segment prefix-free complexity. In
particular, given any computably enumerable set $A$ with non-trivial
prefix-free initial segment complexity, there exists a Turing complete
computably enumerable set $B$ with complexity strictly less than the complexity
of $A$. On the other hand it is known that sets with trivial initial segment
prefix-free complexity are not Turing complete.