At the end of December 2015 Avid introduced Media Composer 8.31 which can edit at resolutions up to 4K. All previous versions, including version 8, can only edit at up to HD resolutions.
Avid also introduced their own 4K codec, DNxHR which can handle 4K file sizes. Avid can output 4K resolutions using Blackmagic hardware.
Avid's 4k playback is not as good as Premiere Pro or Grass Valley EDIUS if you are using native footage (ie footage which has not been converted to Avid's format). However if has more options to drop the resolution of the playback and the output via your hardware than other programs. For example you can switch the output on an MXO2 to output HD from a 4K timeline easily. The proxy modes change the resolution at which the footage is played as well, which means if your system is not capable of playing back 4K at full quality you can drop it to a resolution with which it can cope.
Avid also lets you make 4K Quicktime files in their own format so you can get full quality 4K edits out of the program and into a different program if you need to. The codec for this is free so you can install it on machines that do not have Avid installed. This is similar to Grass Valley and their HQX codec.
Avid's system requirements for 4K editing at quite high. they say you should have a dual 10 core Xeon system as an "entry level" 4K editing suite and ideally a faster computer for more 4K editing. Whilst we would agree you would want as fast a machine as possible these requirements are quite high compared to Premiere or EDIUS. You can see details of the validated HP Xeon systems here: http://www.dvc.uk.com/acatalog/HP-Dual-Xeon-Validated-system.html
Avid Media Composer 7 will load most types of 4K footage but cannot currently edit footage at 4K. The highest resolution at which you can edit is 1920x1080 50/60i. We are hoping this may change with some updates due this year, but we have no indication whether this will happen or not.
To import footage you can either use the standard import feature or Avid AMA. If you use the import feature the footage is scaled to HD on import and the 4K resolution is lost. If you use AMA then you have access to the full resolution of the image and can reframe and pan and scan it using Avid frameflex. The only problem with this is that if you AMA a lot of footage than Avid becomes unresponsive.
In our tests Avid loaded footage from the Blackmagic Production camera and the Panasonic GH4, but would only load the latter type of footage is filmed in MOV format.