Acc. to Rav Shlomo Zalman Auerbach (Minchas Shlomo) there is no problem with a circuit that is broken or ineffective on Shabbos. However, the items are still a Keli Shemelachto l'issur - a vessel that is used for a forbidden melacho (i.e. a microwave oven) cannot be moved or unplugged on Shabbos.
In the case of an implant, if it is a circuit that is currently running and can be turned on or off, that would be an issue. However, if it is like an NFC chip - that is only activated when brought into proximity with another circuit, then, as long as it is not brought into that vicinity then it shouldn't be a problem with regard to completing a circuit.
However, there is an additional issue. The Torah describes the Melachos as Melachos Machsheves. What this means practically is that today you can get lamps that turn on when they 'hear' a human clapping. Now, on Shabbos it is permitted to make a clapping sound (according to the Rama, according to others, (like the Rosh, it is better to use a Shinui and clap with the back of the hands, like the Yerushalmi says). Does that mean one can turn on a lamp with a clap? So the answer is that no - since the intention is to turn the lamp on, then it is forbidden (and if it is an incandescent light bulb - possibly D'Oraissa acc. to the Rambam). So if your intention is to do a forbidden act, even if the implant seems to respond to a simple hand gesture, it would be forbidden.
Aside from that, there is a further issur of causing oneself bodily injury. Many poskim do not allow implants or plastic surgery for the sake of beauty. If there is a medical necessity or some sort of medical reason that is needed, there would be a heter to allow it. It would seem to me that having an implant for non-medical purposes would not necessarily be under this category of allowance.
As regards Cochlear implants and hearing aids, this is precisely what was addressed by Rav Shlomo Zalman Auerbach. As long as one does not adjust it on Shabbos, there should be no reason to forbid it (although some are machmir and turn off their hearing aid on Shabbos). Chochlear implants might even be Hagrama (indirect) since they take sound waves and convert them into neuro-impulses. It isn't a malacha machsheves, because the intention is to hear, and not to do a melacha per se. As regards whether one should have an implant - this is certainly regarded as an enhancement for better quality of life, and will enable someone to hear Torah and be able to learn and speak so there should be no reason to forbid an implant on these grounds.