There is no such thing as a kind of turing completeness in linguistics. natural languages are not even finite and are rarely used to express "computations."
Infinite tape is a requirement for Turing completeness. A whole lot of language is used for making sets of things, and inclusion or exclusion from those sets. Take, for example, your comment. And mine.
Maybe set definitions and memberships aren't computation though. I think they are, but I'm a fuzzy thinker.
