Practically speaking, we already have more than two parties. Both parties have multiple factions. Instead of a parliamentary system where two or more parties must coelesce to form a majority government, we winnow the various factions through a more elaborate system of primary elections, but that really isn't that dissimilar.
"Principled conservatives" (i.e. non-pragmatic) and "moderate Republicans" (i.e. all in favor of big government as long as they're in power) are butthurt because Trump's populist faction won, but that won't always be the case.
Socialists are butthurt that the Dems rigged the primary to defeat their choice, and they have every right to be pissed about that. But they fail to realize that Bernie would lose the general in a McGovern style landslide.