Very interesting - I mean, it actually could be possible he'd do that (unironically), but I still find it very unlikely. It's been known to happen that senators who leave office switch to the other senate seat (Goldwater, most prominently), or people who run for a senate seat and then lose run successfully for the state's other senate seat later on - but for an incumbent senator to run for the state's other senate seat, that'd be very unusual indeed (a first?). I would dismiss it completely, and probably still will, but it's not entirely possible if Menendez becomes a bigger embarrassment and Booker wants to prove something.
Kent Conrad promised in 1986 not to run for re-election if the budget deficit didn't fall, which was moderately stupid on his part. Quentin Burdick, who held the other seat, died, allowing Conrad to run in the special election for the other seat.