A top two primary system was on the November ballot in Arizona and was rejected by the voters (Source).
In a top two primary system, all candidates, regardless of party, are on one ballot, i.e., there is no Republican or Democratic primary - just one primary with all candidates. The top two voter getters will go on to the November election.
The argument in favour of such a system is that it allows more moderate candidates to emerge, as opposed to the trend that appears to occurring in the current primaries where the more extreme candidates may win (Source).
Unfortunately, I have not spent a significant amount of time studying this issue; however, when California passed their version of this system, it was suggested by some that Texas use a top-two primary and I commented on that at the time. Looking at Lousiana, who has used a top-two primary for some time, I found that there were several elections in which two candidates were from the same party.
This seems to be the case in California as well: “Twenty-eight races next month feature two candidates from the same party: 19 Democratic contests, nine Republican. Minor parties are almost completely shut out, surviving in only three races” (Source).
My guess is that you’re not going to see much moderation as a result of this system. Depending on your district, you may see candidates move back to the center after a primary, which is what is happening in some of these elections in California (so not much difference in that regard). When I looked at Louisiana, I did not review voting records to see if any moderation occurred; however, a review of scores from special interests groups could give an idea on that point.
There may be some data/research already existing on this topic and I haven’t found it yet.