I'm more focused on linguistically and culturally, but politically would be interesting to discuss as well.
Let's say for the sake of argument that Quebec joined the American Revolution of its own volition, and gained a plurality or majority of support for the patriot cause. People like Fleury Mesplet, Christophe Pelissier, Denis Viger, Francois-Pierre Cherrier, and others become prominent patriot leaders and Quebec joins and stays in the US to this day. What would this state look like? Does the French spoken here change in any way, vocab-wise, spelling-wise, or pronunciation-wise? How about movies/TV culture?
So far, I'd guess they would likely still have a bicameral legislature, and a governor. The surrounding states would teach French in schools (NY, VT, NH, ME, NS, any eastern/western bordering states). Interstates would have the same signage as in the rest of the country. Movies/TV Shows would come from Hollywood (if that still exists) dubbed in French.
Jacques-Felix Sincennes' northern railroad would let Quebeckers and other northern immigrants spread across the north, possibly making northern states more populous without the international border there.
I would imagine they'd also lionize their local patriots who fought for independence from the UK much like the other states did. Politically, I can imagine the state would industrialize early with the rest of New England, support the Civil War on the Union side, and be Republican early on, until the latter half of the 19th century, and with the rise of unions, likely switch Democrat and stay that way for at least several decades, possibly till the 40s/50s, and become a swing state, if the Democrats and Republicans follow roughly the same evolution. If the Republicans steer more libertarian, that could possibly make Quebec a swing state in the latter 20th century.
Any other thoughts?