The ideological repertoire represented by American Democrats and American Republicans is in constant flux; a “democrat” in 1960 could have vastly different political views than today’s democrat. I think it’s likely that we’re on the cusp of a major political shift, but it’s not clear exactly how it’s going to play out. I see two possibilities.
The first is that libertarianism rises from the “center” and steals voters from each side. I believe there’s already enough closet libertarians out there to outnumber hardline democrats and republicans, but the libertarian party lacks the internal cohesion and the momentum required to breakthrough as a formidable alternative. Gaining that solidarity and numbers would require no less than a highly proficient and charismatic leader to be the “one.”
The second possibility is that both parties will fracture and we’ll either have a four-party system or they will reconglomerate as two new major parties, which may or may not carry along the nomenclature of “democrat” and “republican.”
As of now, the right has been fracturing into establishment republicans (also known, ironically now, as neoconservatives), and the alt-right.
Neoconservativism is a kind of evolved liberalism; they were originally called “liberals mugged by reality.” They believe in free market and civil rights, but they stick out for strongly self-identifying as Christians and patriots, both of which have an impact on their ideology.
The alt-right was born of Internet-age nihilism; they see civilization as a grand and perpetual culture war, and seek refuge in the trenches dug by American paleoconservatives. They believe in one human right above all others: free speech.
Less widely understood is that the left is facing an internal bifurcation as well, and interestingly, free speech plays a role in this one too. The split has to do with the belief held by some leftists that freedom of speech, press, and expression should not be granted to those whose speech can be interpreted as oppressive. The leftists who oppose this asterisk to free-speech like Sargon of Akkad and Mark Lilla are contrasting/distancing themselves from the proponents by calling themselves “liberals” instead of “progressives.”
Psychologist/philosopher Jordan Peterson sees progressives as followers of postmodernism, broad philosophy that (quoting the Wikipedia article) “asserts to varying degrees that claims to knowledge and truth are products of social, historical or political discourses or interpretations, and are therefore contextual or socially constructed.” In other words, gender, sexuality, ethnicity, and religion are all social constructs, and an individual who speaks about the social construct of which they are a majority member is acting oppressively, because he or she is perpetuating his or her dominance in those categories by speaking about it.
To me it seems that the alt-right and progressives would be unlikely to ever form an alliance, because the alt-right is, in a lot of ways, a defensive reaction to identity politics of progressivism. Liberals and the alt-right have a little more in common, given that they hold the first amendment in high regard, but there would be a lot more issues they would have to reconcile before cooperating. The same essentially goes for a liberal-neocon super-party of moderates; the two would have to agree on a number of social issues like gay marriage, legalization of marijuana, and immigration, but they might more easily come to a consensus on regulation and globalization. I therefore maintain that the most likely shift in the American political landscape would be the rise of a libertarian-minded third party.