TalkChess.com

I thought that it is possible to give rating of complexity for chess positions based on the gain that strong engines get from doubling the time control from 2 minutes/40 move to 4 minutes/40 moves.

The idea is that you decide for example that the top 50 engines in the CCRL rating list participate and make a tournament when every engine has 2 entries:

enrty 1:the engine at time control of 4 minutes/40 moves
entry 2:the engine at time control of 2 minutes/40 moves

100 players can play 198 games because every possible pair play one game with white and one game with black.
You calculate based on the tournament the difference in points between the entries that used 4 minutes/40 moves and the entries that used 2 minutes/40 moves and you have a score for the complexity of the position.

Obvious drawn position or obvious win for one side of course are going to get rating of 0.

I do not know if it is possible to generate a position with negative rating but I guess that it may be possible if there is a trap that mainly strong engines fall into it and if you do not fall into the trap the result is obvious and finding the right moves after an engine fall into the trap is also relatively easy.

Note that my initial thinking was to calculate rating for all engines and look at the average rating difference but I decided that it may be impossible because some engine at 2 minutes/40 moves may score 0 points in some composed position and with 0 points it is impossible to give rating when it is possible always to calculate average of number of points the engines get.

I remember reading an article around 20 years ago stating that in tactically complicated positions, chess programs are not able to search as deeply (in terms of ply) as they are in simpler positions in the same amount of time. Would Uri's measure produce different results to this?

I don't know, but I can confirm just looking at the depth that the engine reaches is a good way to measure complexity. Some positions are so complex that it takes me 1 hour to reach depth 32, while others are so simple I can reach depth 60 in 10 minutes. Interestingly, this can be used to defeat weak opposition, specially positions that are outliers (they're very complex, yet the engine reaches high depth quickly, because it's examining the wrong lines; once it sees the right line it can spend most of its analysis time stuck at this depth) seem very difficult to play for some people, for some reason.

Another less reliable method is looking at nodes per second, with higher indicating more simplicity, but this would only be useful when comparing positions with comparable depth speed.