Twitoaster takes conversation tracking to new heights for Twitter. Other sites and products would just try and show some form of reply hierarchy to tweets but it was often incorrect or still hard to follow.
You log in with your Twitter credentials into their web interface (no desktop client). Hot conversations in all languages (it shows conversation counts) are listed on the lower part of the page for all to see before logging in, by the way.
One logged in there is a welcome page telling you that it is loading your conversations for the first time and to wait a second or follow them and they will DM you when done. I am not sure how far back it goes and a lot depends on how much you conversate (as I promised to do more).
Your stats and ranks say they will take a day to show up, but it pulls your Twitter bio immediately.
You can see overall rankings with a tab at the top of the page and I was impressed with the charting ability built in. They have embed code for each of the charts for your blog or website to show how interactive you are. I really liked that feature. Here is a sample from me :
So I waited part of the day and revisited and my conversations were in. Clicking the number on the tweet shows the threaded conversation in a drop down. Very slick.
Clicking on see the whole conversation allows you to reply, share across social networks and even retweet. This is a site to use!