I like that Hamlet (potential) tie-in, it has an elegance to it that works for me.
Whether or not that's correct, I understand the "I" to be Hal's consciousness (or soul), his "real" non-repressed/buried self, which has been silenced/unable to communicate with the outside world for whatever reason. We see in the YOG section that Hal is able to think, remember, reason normally, but he cannot "interface" with third parties; he is unable to make "I"-contact with others (forgive the pun). That's a good catch on the "what he sees as my eye" line -- I think that ties in, remembering that the "eyes are the window to the soul."
To put it another way, by this point, Hal's consciousness/soul is completely abstracted from/trapped within the "machine" of his highly-skilled body. The administrators rightly worry they would be viewed as "using" Hal (really, his tennis-playing machine), while Hal's "I"/consciousness internally protests "I am not just a machine." He is, literally, the dues ex machina -- the "ghost" trapped in the machine of his body.
As for how/why Hal's consciousness gets abstracted from / trapped within his body...
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