I disagree with the concept to abolish the political parties. While party partisanship appears to be a major problem right now, I feel that is actually a smokescreen. In fact, it is possible that strong political parties are needed now more than ever, an idea I'll explain a little further down.
The party system historically has worked as a buffer against extreme ideals and has actually helped elective offices remain more balanced than more dissimilar. Without a party system, it is highly more possible that candidates with strong, extremist viewpoints could get elected. At the root of this was funding - it has been nearly impossible for a candidate to secure office without the backing of his or her party. With a majority of party finances being derived from donors and sources that are somewhat more centrist, no political party would be in a position to totally disenfranchise their base by swaying too far from a reasonable position.
However, one thing has significantly distorted this - the Super PACs. With vast sums of political funding now driven by Super PACs, fringe candidates are more easily able to mount viable campaigns without the express approval and support of their party. People who were once not electable are now viable candidates, and they are occasionally elected now and are able to buck their party's agenda.
Another aggravating factor is how the use of media has shifted over the last decade or so and how that is shaping the electability of candidates. At one time, the various media outlets were also somewhat more centrist in their approach. There were certainly biases present in reporting, but they were much more subtle than what we see today. With news organizations themselves representing positions far more left or right than previously, they shape the viewer or reader's perspectives on the candidates far more than before. At the same time, the voting populace has themselves shifted further away from a population that may actually do some research and create an informed opinion; it is now a world of tweets and soundbites the defines what directions a voter heads.
I recall from college Poly Sci that "An informed and intelligent electorate is essential for a functional democracy". The political parties used to help support that vision by having a platform that people could understand. Campaign funding was consistent with that perception, and helped position candidates who represented views that would reflect that vision but yet be sufficient middle-of-the-road so as to be electable. And the media reasonably presented those opinions to the public to make a choice. However, we largely have an electorate now that is neither informed nor intelligent (at least on the fundamental issues and true needs).
With the terribly skewed Super PAC-based funding and the shifts in relatively balanced media coverage, we now have a system where the loudest candidate voice gets the most sound bites and is packaged and marketed in ways we never saw a couple decades ago. The extremists now have a platform, and they have found the self-perpetuating cycle where the louder they yell the more funding they can generate. Chris Anderson's book "The Long Tail" outlined a business phenomena where smaller demographic segments offered great opportunities through technology-infused marketing focus, and this same phenomena can now be seen in our political spectrum. The "long tail" demographics in the electorate have power unlike they ever had before. Without the two main political parties providing some buffer in the system, it is likely this problem will get worse and worse.