Quite the salient points from @LordDemiurge, but because this one's particularly germane to my own take:
Little fanciful and "spoiler-ish", but in my latest outline, Neo-Augustus unironically assumes the title of "Lord Executive". In large part, it's a reaction to how "POTUS" was disgraced by Neo-Sulla's excesses — and consequently, fell out of favor in much the same way that "Consul" lost its prestige with the demise of the Roman Republic.
But the other (and more important) component to this is how my take on Imperial America is more of a voluntary, pan-Western mercantile league with various parallels to the Hanseatic example. That is, an easygoing confederation in which a smorgasbord of locales, mini-states, and “free zones” assume a voluntary membership where they agree to provide payment (and perhaps, manpower) in exchange for common defense, free trade and travel, and general adherence to the Lord Executive's new and improved Constitution.
Granted, it funds quite a few agencies and functions the original Hanseatic League did not: a league-wide military, treasury, and administrative directorate, for starters. Otherwise, it's very "transactional" and business-minded in its outlook, which even extends to the various titles it grants ranking officials and the prominence of American merchant families in its middle and upper echelons. Hence, the head-of-state as "Lord Executive", in the sense of Neo-Augustus (and of course, his successors) being the CEO and public face of the American mercantile league (more or less).
Indeed, a distinctly "American" approach to becoming the Mercantile Empire of the West, given the US's distinct entrepreneurial spirit and tendency towards commercial ventures as it stands. To paraphrase Calvin Coolidge: "The chief business of the American people is business." — an adage that Neo-Augustus (or at least, the version who emerges on top in my outline) very much takes to heart.
That would be interesting. Makes me think of the early years of the Delian League although whether it would then fairly quickly convert to a more explicit American empire as the Athenian version did. The Hanseatic League was more of an 'alliance' of at least roughly equal power rather than with one state being/becoming overwhelmingly dominant as in the Athenian case. I think most posters are thinking of a much more centralized power system such as Augustus set up and then somehow it becoming a more decentralized feudal medieval system rather than the growing centralization of Rome until it collapsed under its own weight.
Anyway the rugby is starting so my attention will be elsewhere for a couple of hours.