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.
Since this a thread about Menendez, there's a sort of similar example that happened in a New Jersey legislative race in 2019.
In Morris County's LD25, long-serving State Senator Anthony Bucco died of a heart attack in September 2019. His son, Assemblyman Tony Bucco, was appointed to replace him in the Senate.
The problem is that the younger Bucco was already running for reelection to his seat in the Assembly and because the appointment happened so close to the November election, it was too late to replace him on the ballot. Effectively, he was running for an office that he had no intention of actually holding.
Despite all that, the younger Bucco still won the Assembly race and resigned shortly after to officially take the Senate seat.
