As shown, date complications are very tough to calculate and there are many levels of features.
However, the phases of the moon complication is not discussed in much depth here. Is it a simple calculation independent of the date?
Many watches use 29.5 days between phases for simplicity. This results in the watch's moon phase display being off by one day for every two years the watch operates.
There are a few watch manufacturers that use epicyclic gear trains to make the moon phase calculation more precise. An example: the Ochs und Junior moon phase watch will operate for 3,478.27 years before its moon phase display is off by one day (https://www.ochsundjunior.swiss/watches/moon-phase/).