Is there an alternative cheap or free Twitter app that supports Retina display?
There are several complicated and time-limited things that are causing apps like Tweetbot to very likely get all shut down by Twitter in the near future.
My impression is the high price of Tweetbot not only reflects the high cost that the developers spent to make such a complicated and design intensive app, but also to discourage too many people from "trying" it out as well as a calculation that they need to make their investment back sooner rather than later due to the uncertainty of being dependent on Twitter's API rule changes.
Twitter is in the process of shutting down all third party apps that replicate what you see at http://twitter.com in terms of showing you the timeline.
In a nutshell - in the last 12 months, Twitter has been telling developers to not make apps like Tweetbot since they replicate the core twitter experience and the 100,000 user token limit is really making it impossible for a team of talented professionals to recoup their costs without charging for an app since you need way more than 100,000 users to make money on data or advertising.
Twitter wants to make money, developers want to make money, so at the moment the development of great, free twitter apps isn't exactly a hotbed of innovation in 2012 like it was in the past.
Many in the development community had assumed that Twitter would bend the rules when their app wasn't ready, but with the Windows 8 release, they shut down the most popular client when it hit 100,000 user-tokens even though Twitter doesn't even have a windows 8 client (perhaps yet). When that happened, I imagine a lot of projects are being dropped or at least halted. Not only is Twitter putting hard limits on third party app growth, it also is getting more and more exacting in telling these devs exactly how a tweet must be shown (with the implied threat that they will shut down those apps if they fail to comply to Twitter's exact design standards which include serving all promotional content)
So, now that a brief back story is complete, here is your answer:
The following OS X twitter clients have been viewed to broadly and/or fully support retina display Macs:
Both Hibari and Twitterrific have free downloads so you can try them both before buying. Check all of them as one is on sale now for half price and prices can change over time. All of the above render text with great precision and have all or many of the app's graphics in retina. Of course the twitter stream has many non-retina people thumbnails, etc...