[vurpo]

Actually, how would you do a syscall in Brainfuck? Its only way to interact with the outside world is through "input byte" and "output byte", and that does not give you the ability to do arbitrary syscalls. As far as I can tell, arbitrary syscalls are impossible to do in Brainfuck (which is why that one person made a fork of Brainfuck with added syscall support).

Of course, syscalls are not related to Turing-completeness in any way. Turing-completeness means that the language can express any computation which is possible to do on a Turing machine. Turing machine computations can not have any side effects, and syscalls are a kind of side effect. Therefore, having syscalls is not necessary for a language to be Turing-complete.