My Raynaud's hands do that, but to be fair I have other circulation problems, too. I have some form of dysautonomia, and a connective tissue problem, and ME itself could impair microcirculation in some way, which could lead to this result. I actually get discoloration extending up my limbs. (That's why I like my PCP: because he notices stuff like this and it matters to him, even if he does not know what it is exactly.)
One thing you can do to test how bad it is is a capillary refill test (press on the end of a finger until it turns white, and time with a stopwatch how long it takes to turn pink or, well, purple will do, again). It should be under 2 seconds. If it is less than 2 seconds, make sure your doctor notices next time you are in and have discolored hands. (If you are like me, it is not hard to get while you are out and uncomfortable.) Go ahead and send your photo to your doc now, if you think the doc wouldn't mind.
If it is more than 2 seconds, I would suggest to call your doctor or go to an immediate care center.
This is a capillary refill test and is described here:
https://www.nlm.nih.gov/medlineplus/ency/article/003394.htm
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4513433/
The timing is the same in pediatric:
http://www.ncbi.nlm.nih.gov/pubmed/21238980
Of course, I'm a fellow patient, not a doctor, nurse, etc.