I was a bit confiused, as an ENTP, being an Ne-dom, would be brimming with ideas, while your comment was indicative of someone who was waiting for someone else to make the decisions for him. So I looked up quotes for Barbarossa, and recalled the films, which I had watched several times. He seemed to be pretty focussed on what practically concerned him and his crew. He also seemed to relish taking charge. He also seemed to be very uninterested in the theory of how and why he and his crew became ghosts. He seemed to be only interested in becoming human again. So he seemed to lack a lot of the characteristics of INTPs, and seemed to have a fair bit of the characteristics of a Te-dom. So I thought that ENTJ was more likely than INTP.
Well, I could look at the whole thing from scratch.
Barbarossa is definitely a take-charge sort of guy, and usually has a plan, when he wants to do something. Just look at how he handled the British in the first film. Doesn't let on about his plan to have his crew walk under water, and take the British, using their immortality as ghosts. So I'd say that he has to be a J.
He seems to be more interested in what practically concerns him, which suggests T. But then again, he seems to mostly be interested in fulfilling his own desires. So I can see both sides there.
Definitely not an Idealist, though. So definitely not an xFJ. So he has to be an xTJ.
He seems to keep his own counsel, using his crew as an extension of himself. So on reflection, he could be an I. But way too smiley, to be an INTJ. Way too inventive in his plans and his communications, to be an ISTJ. So I'd say ENTJ, or possibly ESTJ. I'd favour ENTJ, because he likes to challenge people so much.
Probably why he can follow what Sparrow says, but argues with him so much. ENTPs and INTJs normally get on, because they are both Intuitive dominants. ExTJs and ENTPs are both extroverts. But their thinking is driven by entirely different perspectives. The Te-dom is driven by what is useful to him, while the Ne-dom is driven by the possibilities that can be achieved.