Although there is rampant disappointment with our two-party political system in almost all directions, I don't concur that the two-party system is either "doomed" or even "dysfunctional." The essence of our current system is that forward progress occurs when there is consensus and the status quo is maintained when there is not consensus. When parties are too far apart, we simply wait until conditions change and enough compromise occurs to reach a deal. That's the way America has always worked and continues to work.
The fact that some dislike or even loathe compromise is a statement about themselves and their own values rather than an indictment of our underlying political system.
The fact that we may not be moving forward on every front as fast as partisan political interests may desire is less about "dysfunction" and more about individuals contriving false expectations in their own minds, albeit often with the assistance of the media and partisan political commentators who thrill to the "fight" and excitement of partisan political squabbling than to the more mundane details of compromise.
The essential problem for many people is that they feel that their voices are not being heard when they "compromise" and toe the party line. That's why I believe that we need a four-party system. Not because the current system is "broken" (it's not), but to evolve to a system where more people feel more comfortable and satisfied with our system.
My basic proposal is to essentially split the two dominant political parties into two parts, the centrist and moderate wing, and the ideological and activist wing. The basic thesis is that the vast majority of Americans can easily and readily tolerate the moderate wing of either party, so that if we bounce back and forth between the moderate liberals and the moderate conservatives, the vast majority of average Americans will still be able to sleep well at night. Liberal and conservative activists may not be too happy about that arrangement, but my thesis is that having their own parties will leave them much more satisfied that their voices will be heard.
My thesis is that for any typical election the two moderate parties will compete for votes from the opposing moderate party and their own activist wing. Sometimes they will be able to muster enough votes from the latter, but typically the extreme ideologies of the activist wings will tend to lead the moderates to come to a compromise more often than not. In other words, sometimes the activists and moderates would come together, while sometimes the two moderate wings would come together.
At the presidential level, moderate compromise would be the norm. In extreme cases an activist wing might lead the charge, but more often "common sense compromise" would rule the day even as activists may seethe in opposition. Sometimes the moderate plus activist wing will be sufficient to carry the day, but more commonly one of the moderate wings would decide to throw its support to the opposing moderate camp in order to further marginalize the opposing ideological activists.
At the congressional level, individual districts or states could well elect a number of fringe or activist candidates, although they would tend to caucus with the moderate party closest to their ideology. Even if most ideological activists can't agree to compromise with any moderates, it is far more likely that the moderates would compromise with moderates of the other party. The upside is that even if the ideologues don't compromise, they gain the benefit of having their public voice without distorting the values of their moderate colleagues. They gain the opportunity to have greater influence, by sacrificing the privilege of membership in the moderate party.
It would basically be a win-win (or win-win-win-win) for all parties.