Hi, many have solved the problem using any of the thunderbolt adapters, but not all...
I can't find it now, but just read one that had to use TB->DVI->DVI->HDMI!???
I had better luck with a Mini DisplayPort to DisplayPort adapter. The picture was much better, with HDMI it looks awful (this was before my 2012 mini died after 3 weeks). However, when I hooked my 2011 mini back up using MDP to DP I was shocked at how much better the picture quality is. The colors are sharper/brighter and the text is better. The people at the Apple Genius Bar I spoke to claim they've never heard of any issues with 2012 mini's and picture quality problems (c'mon).
I have read the various posts concerning the ongoing HDMI issues with the 2012 Mac Mini when attached to various monitors.I'm using a HDMI to HDMI cable on a Mac Mini and the colors look white washed, example you open up a folder in list view you can not see the light blue colored rows that should appear. Text also looks very blurry / jagged.
Go the Apple > System Preferences > Accessibility > Display and make sure that Enhance Contrast: is set to Normal.
I had exactly the same problem, but with a MacBook Pro, and in my case it was an easy fix. OS X thinks my monitor is a TV, and is using the YCbCr colour space rather than RGB. I had to override an EDID setting to force the RGB colour space, and it is now working correctly.
Here's how to do it:
I'm using the included HDMI-DVI adapter to connect to to my old Samsung SyncMaster 244T and the ouput is fine - no crushed whites, fuzzy text, etc. My guess is that there are still some problems with 10.8.2 forcing the wrong profile/color space onto the monitor when using HDMI-only - hopefully 10.8.3 will fix that.
See also these threads about the profile/color space theory: