The watch has the most amazing hidden display I have seen so far. I have an Omega Seamaster Multifunction (Tissot Twotimer module by SSIH used) which also hides that digital display nicely. But this one is really amazing. By looking at the watch in closed state, you would never guess that there is world time, day and date, chronograph, alarm and timer hidden...
The analog and digital displays are not synchronized, but on the flight while non-stop playing with the functions I noticed that the separation of the circuits also has an advantage. By separating the analog and digital circuits, one can very easily switch the time zone and can be sure that one does not lose a second. If you are not a WIS, this might have no meaning for you, so do not worry too much about those explanations.
The IWC Caliber 630, also known as Mecha-Quartz, is another watch that features an independent chronograph (you start it when the main seconds reach the 00 position and remember the hours and minutes you are going to set) that helps to adjust main time easily. Of course the Lange Saxomat with push-the-button-synchronization belongs into this category too, but at a slightly higher price :-)
The easiest world time watch is my Casio Mr.G Titanium Analog Digital watch. One push of the button will exchange home versus world time and the digital and analog circuits are synchronized.
I will later add links to the watches mentioned, but for now, please see those watches on their pages in my private watch collection. And ... you will not be finding the Saxomat there, I do not own a Lange watch :-)