In 1987 (yeah, I'm a geezer) I bought a JL "Lyre," which was a complicated watch. The damn thing never worked properly and a year later JL took it back and gave me an "Odysseus." (My first Lemon Law case I suppose.) Here's what it looks like. (Mine has rose gold on the case.)
It's also a complicated but has one more complication -- the 24 hour clock on the inside part of the dial where the second, minute and hour hands emanate from -- it is marked in red numbers from 10pm to 3 am -- that is the time when you must not seek to adjust the watch. (The Lyre had everything except the 24 hour clock.)
It cost $17,000 in 1988, and today a similar line of such JL watches is about twice that.
Note that the Odysseus has only the one "crown" (winder). All of the functions are accessible from there. But if you set it ahead you have to wait until that date to let it start again. And you must never move anything from the crown from 10pm to 3am.
The watch will keep time and tell you the phases of the moon, and do leap years until March 1, 2100. I've made an appointment at my jewelers to go in then for the adjustment. (Hey, why not? My Dad is 99 and in great health.)
Jaeger LeCoultre is a great brand and watchmaker. They make the movements for everybody other than Rolex and Patek Philippe. So you've made a good choice.