Yes, you will be able to run Adobe Cloud on multiple platforms provided that you will only use Windows-based OS and/or iOS. It will be able to run on different computers as long as it is compatible. You can actually use your Adobe Cloud account on two different computers. You can already imagine how convenient this will be, especially when you are going to use one at home while the other one can be used in the office.
There are also some required system requirements so that you can be sure that Adobe Cloud will run well on your computer. You need to have enough RAM memory, a fast processor, and a video processor that will be able to take the needs of Adobe Cloud.