I don't want to give a "don't do it" answer, but I would say that it's difficult, so it'd only be worth trying to negotiate a PhD, academic publishing, where you fit into a discipline, etc., if you're really committed to a research career. It also depends on what exactly you'd want to study; a lot depends on finding a supportive advisor who would be willing to supervise the kind of thesis you want to work on. This depends not only on the style of work, but also the specific domain, e.g. you're going to get a totally different set of candidates if you're interested in, say, interactive entertainment (there's a sub-field of game AI, AI-for-narrative, etc.) or perhaps something to do with robotics (also its own subfield), or else something to do with human-computer interactions (something vaguely in HCI, CSCW, etc.).
"Big AI" isn't very much in favor currently, partly for good reasons and partly for bad reasons. There's a strong worry about being too unrigorous or philosophical or vague or even sci-fi. Academic AI probably overcorrects for a fear of being seen like crazy singularity-mongers, and there's also a legacy of having over-promised some big-AI stuff in the '50s and '60s. Most funding is also for more concrete technical projects, though there is a subset of people doing some funded research in the area of artificial creativity and creativity support (Margaret Boden and Gerhard Fischer are two entry points into that literature).
So, most research tends to be much narrower and investigate specific empirical or mathematical questions, like whether a particular reinforcement-learning algorithm converges, or how to improve an object-tracking algorithm, or something of that sort. Even in Cog Sci departments, the theses tend to be more specific, like doing an eye-tracking study that investigates some question about perception. To the extent the "bigger picture" stuff gets done at all, there's a feeling that it's a late-career thing people like Hofstadter can get away with, but it's harder to do as a PhD thesis.
Not sure that actually answered your question, but the short version is: it's hard to get in a position where you can study the kind of stuff discussed in GEB, but if you can think of more specific technical questions on the peripheries of your big-picture interests, it may be more doable.
"Big AI" isn't very much in favor currently, partly for good reasons and partly for bad reasons. There's a strong worry about being too unrigorous or philosophical or vague or even sci-fi. Academic AI probably overcorrects for a fear of being seen like crazy singularity-mongers, and there's also a legacy of having over-promised some big-AI stuff in the '50s and '60s. Most funding is also for more concrete technical projects, though there is a subset of people doing some funded research in the area of artificial creativity and creativity support (Margaret Boden and Gerhard Fischer are two entry points into that literature).
So, most research tends to be much narrower and investigate specific empirical or mathematical questions, like whether a particular reinforcement-learning algorithm converges, or how to improve an object-tracking algorithm, or something of that sort. Even in Cog Sci departments, the theses tend to be more specific, like doing an eye-tracking study that investigates some question about perception. To the extent the "bigger picture" stuff gets done at all, there's a feeling that it's a late-career thing people like Hofstadter can get away with, but it's harder to do as a PhD thesis.
Not sure that actually answered your question, but the short version is: it's hard to get in a position where you can study the kind of stuff discussed in GEB, but if you can think of more specific technical questions on the peripheries of your big-picture interests, it may be more doable.