Present moment, I don't think there are any equivalent for the RX100iii, but frankly speaking... whether it is the best Point and shoot out there at the moment? It all depends on what you are looking at.
If you want hotshoe and flippy screen... RX100ii is the way to go. Or even the Canon G1XmkII. If you need faster lens... and don't mind it lacking a hotshoe for external flash... the RX100iii is the way to go... it come with its own viewfinder too.
The RX100ii and RX100iii shared the same sensor, and so IQ should be about the same. G1XmkII although having bigger sensor and lower resolution, doesn't win too much or at all with the dynamic range and low light performance as compared to the Sony RX100ii/ iii, which is kind of a surprise actually. But that is not to say that G1XmkII is a lousier camera.
As to people mentioning that Sony picture looked flat, it is also quite subjective. I find the images (from reviews site) from RX100iii quite pleasing, no issues there and it certainly don't looked flat. And if you want 3D... then all these cameras are not for you.
Anyway, there are rumors of a Panasonic LX8 or whatever LX that is sporting a 1 inch sensor also. Can wait for that camera to come out before making a decision. Also, you can check on Samsung NX mini, which is very small and having great 1 inch sensor that had a very reputable IQ... and at a very affordable price.