No, the human brain is not Turing complete. In fact, computers aren't Turing complete either. Turing completeness is the ability to simulate any Turing machine, and Turing machines have <i>infinite</i> memory. The brain has only finite memory, therefore, it cannot simulate a general Turing machine.<p>If you mean "Turing complete" in the sense of "able to run the operations of some Turing complete abstract computation system", the answer is yes. If you can comprehend the instructions of a standard programming language, then by definition you are able to "execute" them in your mind, so you can "run" programs as long as they don't exceed your memory limits – just like any physical computer. The "human brain interface technologies" required already exist: Books that teach you how programming languages work.
No, it's a finite-state machine. However, it becomes Turing complete if you provide the human with a virtually unlimited supply of pen and paper. Richard Feynman once said about his notebooks that "They aren’t a record of my thinking process. They are my thinking process. I actually did the work on the paper."
In the sense that computer peeps use the term, yes we are already - because we can construct and run Turing machines in the real world with paper and pencils. No the brain interface you were going for, but that's my answer.