That bike has some features that make it stand out. It's a single pivot, the shock mounts to a truss between the top tube and down tube, it's a linkage driven and the (missing) linkage attaches to the top tube. The curve to the top tube is quite unique too.
It appears you have a Cannondale Perp, circa 2007-2009.
Some quick research show it shares some parts with the Judge, like the derailleur hanger.
Between the missing hanger and brake dropout you are talking major money to restore. The missing linkage and associated bushings etc is a major issue unless you have them in a box somewhere. I assume finding a compatible shock isn't a huge deal. Jordan "Boostmaster" Olthuis has been able to find shock for his old Norco Shore and it's probably a similar size required.
Luckily Cannondate still has the manual available if you want to find the part numbers.
If this model bike is nostalgic to you, you can likely find a complete bike much cheaper than finding parts for this frame (again, assuming the linkage and dropouts are missing). You would have an easier time sourcing parts for a more collectible brand and model bike if you are into vintage downhill bikes.
Here is a pic of the 2008 model taken from the 2008 Cannondale catalogue (also available on Cannondale's website):