Here's the thing with moon phase complications: they're popular, they're mechanically simple, and they're very bad at accurately displaying anything other than full and crescent moons.
The way most work is to have a semicircular opening with a smaller semicircle, or occulation disk on either end of the semicircle. As the moon emerges over left side, it starts as a tight crescent, and slowly grows to a full moon, before waxing. But the middle areas between the full moon and crescents are poor representations of the moon. If you check your moon phase at a half moon, it will look like a moon cookie with a bite taken out of it more than a half moon.
Enter Audemars. Their stated goal with this patent was to create a complication that did not add substantially to the complexity of the movement, while providing a more accurate representation of the moon at any given phase. What they've come up with is a really clever mechanism that was patented in the United States on July 30, 2013.
They maintain a few staples of more common moon phase complications. As shown in the patent, it keeps the semicircular opening for the complication, a moon disk with two moons opposed at 180 degrees from one another, and the traditional rotation rate of 29.5 days per half-turn. That is where the similarities end.
Instead of having semicircular occulation disks, the patented complication has two rotatable occulation disks that look vaguely like those three winged boomerangs. The disks can be calibrated to provide more accurate occulation of the moon at particular parts of its orbit. As the moon disk rotates it engages gears that move the occulation disks, typically in one quick movement, like with a date change.
In the end, this is a very interesting, but very high end complication. It adds a lot of pieces to what is typically a simple and inexpensive addition to a watch. I would really love to see this make its way into a watch, either by AP or licensed by someone else.
In late November, 2016, Rolex was granted a patent for a complication that would serve a similar purpose to a chronograph, but with fewer parts, thereby making it more reliable, smaller, and lighter than traditional mechanical chronograph movements.
The basic premise is to have a your normal hour, minute, and second hands (shown as a small seconds in the patent drawings), and then "stored" hours, minutes, and seconds hands. Both sets move in sync until you freeze the "stored" time hands. Then, instead of having a readout of how much time has elapsed as with a typical chronograph, you simply have a display of when you stopped the clock.
Two heart cams then serve to resynchronize the seconds with seconds, and hours and minutes with the actual time. Heart cams are routinely used to zero out chronographs, so they have a long history in watchmaking.
I feel that it's unlikely this makes it into a Rolex watch. (The discussion of why you patent something if you're never going to sell it is for another day.) It's not clear where it would fit in their line, and it seems like most people would correctly view it as a cheaper, less convenient version of a chronograph. It's easy to read elapsed time. It's harder to do watch-math -- It's 8:37 now and I stopped the watch at 7:18. . . It's an hour and nineteen minutes, but it took you longer to get there than looking at a chronograph.
One thing that's notable about this patent is that it relates to a new complication, rather than improvements in materials or reliability to existing components. It's a little out of the ordinary for Rolex.