Hello Dear and thanks for the query
Purple discoloration is not a good sign for the healthy healing. The swelling if the part is not allowing blood to supply the distal part of the finger. I think there is formation of abscess
at the site. You need a immediate course of effective antibiotic and anti-inflammatory drugs. An Incision and drainage
procedure can be done if your treating doctor feel the collection if pus is more. The material can be send for culture examination which helps in selecting effective antibiotic.
So consult your doctor asap and start the treatment. Sorry to say but any delay in the treatment can cause loss of finger as it is already purple.
Best Wishes and hope for the best