I have seen electric guitars and basses that don't have any pick ups. For instance, here is an image of two Line 6 electric basses that have no pick ups:
How do these instruments work if they have no pickups?
Piezo-electric transducers lurking under the saddles on the bridge! just like acoustic guitars have possessed for ages. Now bassists have the opportunity to use this technology. It's not new, but quite new on basses. Next may be a 'hybrid' with standard pups and p-e ts.
As well as the in-bridge piezo pickups (either for acoustic tones or MIDI, which is what the Variax guitars use to feed the modelling system), there is a relatively rare second option: hidden pickups. The "Type I" Fender Marauder, for example, had magnetic pickups concealed under the pick guard: