Any function on a watch is called a complication. The more a watch does, other than tell you the time, the more ‘complications’ it is said to have.
Useful complications are features such as alarm, chronograph, day and date, a second timezone - these are practical things you might find very useful in everyday life.
However there are plenty of interesting complications, some of which have absolutely no use in the day to day world. Ironically these complications can indeed be extremely complicated.
Just to keep you on your toes, next to none or very brief instructions are ever printed to accompany your watch. Watch designers say it is the job of the retailer to instruct the customer on how to use their watch. Of course if a watch is a gift, or secondhand this is of no use. Google is helping more watch owners than any retailer. You might be needing a very exact page on google to find out how to work and read these complications;
Foudroyante - A sub-dial indicating fractions of a second. Otherwise known as a jumping or flying second hand. If that quarter of a second is important you, you should probably get one. Otherwise, not sure how useful you’ll actually find this.
Dead seconds - Instead of smoothly gliding between the seconds this second hand discreetly jumps from second to second. As a function this seems simple but its the complex build mechanically that this requires that makes this a sort after, if somewhat unnecessary, complication.
Planetarium - Some or all of the planets in our solar system are taken into account and and their relative positions are shown as they revolve around the sun on a dial. Whilst its interesting to know the position of Mars or Jupiter, its not especially useful other than for general trivia knowledge. Its a little vague in accuracy too. Plus, of course if your watch ever stopped and was fixed and restarted than chances are all the planets would be out of line anyway.
Equation of time - This complication displays the difference between solar time and ‘civilian’ time. Due to the non perfect orbit of the earth around the sun, this is always plus or minus 15 minutes. Fascinating in a true geek kind of way, its not going to make any difference to how time runs here on earth. Unless you are Dr Who.
Moon phase - One of the earliest known complications, this displays the moon phases as it waxes and wanes. Typically keeping a 29.5 day schedule which is close to the actual moon, these are promoted as being correct for 120 years or more. Truthfully ideally needing checking and/or adjusting every three years. There is no real great use in day to day life, unless you fish or farm in your luxury watch and even then, you probably have a technically accurate app for that. Or - you just look up at the moon…. Its what the Watch world call an ‘emotional complication’, having no real purpose other than being charming.
Tourbillion - Half the horology world say this is not technically a complication , as it doesn't actually add any functionality. Way back when it was invented in the late 18th century, the tour billion was created as a cage including the balance wheel and associated party around on their own axis. It was designed to theoretically help the pocket watches stay more accurate by helping with the effect of gravity where they were stored vertically in pockets. In the 20th century they were added to wristwatches, almost as an experiment to see if they increased the accuracy. There is absolutely no evidence that a tourbillion watch is any better than a well made and regular watch. That said, they are quite tricky to assembly and look truly stunning. People pay an awful lot for them, despite adding no utility to a watch.