If it helps ease your worries, here's another explanation which you can discuss with your doctor: I once got a false positive... My doc told me it was due to a cyst on my ovary. Apparently, an ovarian cyst emits a chemical that can trick a home test. And if you read the paper insert that comes with the HPT, it does say that the presence of a cyst will produce inaccurate results.
A cysts on your ovary are quite common, especially as you get older. Cysts naturally form as part of your menstral cycle (to release the egg). Sometimes, they fail to pop on time... Most eventually do burst or reabsorb without intervention, although I've heard its super painful when they burst. Mine grew to be bigger than a quarter (smaller than a half dollar) and had to be removed. Personnally, I suspect that the birthcontrol I was on at the time may have contributed to the cyst; as in, that is how my body responds to that type of birthcontrol (by growing super huge ovarian cysts). Again, no medical support for my suspicion; its my own gut feeling (because I discovered the cyst when I was on that BCP and then stopped it, the cyst stay relatively the same size for a month or so, then my doc told me to use start again in the hopes that the PBC would cause the cyst to reabsorb--and it doubled in size). Not sure when you started your Implanon and whether you ever experienced a cyst on your ovary before, but your body could always respond to the chemical birthcontrol hormones in various and unexpected ways. Something to consider.