I have a bruise in my left wrist by my thumb that won't go away. Could this be something serious?

Possibly. A bruise anywhere that doesn't get better with time should raise suspicion for something more sinister than a simple bruise. Although it might be nothing more than something simple like de quervain's tenosynovitis, see a hand specialists for a complete evaluation.
Bruising. If bruising has not resolved after couple weeks or associated w pain and affecting mobility seek professional medical care.