Is there an objective measure of how "complicated" or "tension-filled" a chess position can be? Probably most chess players will have the intuitive notion of what kind of positions are not "tension-filled" and what kind of positions are. Usually these are middlegame positions.

Here's an idea of what kind of positions are "tension-filled": In the position there are many pieces that are either hanging or, many times, a piece of lower material value attacks a piece of higher material value (which may or not be defended). Perhaps it could be defined in such a way as "the total value of hanging pieces plus the total value of defended pieces which are attacked by pieces of inferior material value minus the material value of the pieces of inferior material value attacking the defended pieces", or something like this. This is just a "materialistic" attempt to define the concept, since it doesn't incorporate advantages or disadvantages due to positional factors. Perhaps one could consider Tal vs Hecht, 1962 Varna Olympiad at move 21 as an example of a complicated position.

Nonetheless, is there actually an "objective" or "mathematical" way to quantify this idea which can distinguish a position which is not so complicated from one which most chess players would intuitively agree is "complicated"? Has this been done before?
To me, a position looks "tension-filled" only when there are many threats on the board and (the important part!) - when these threats are mutual. A position that contains many threats but all of them being made by one player - that's rather realization of huge positional advantage.

I don't know very well the quality notation of moves in English chess books, but in Russian ones some moves get the '!?' quality notation, which means "this is a slightly risky move that can give you advantage if your opponent fails to find the right countermove (which is not trivial to find, that's the whole point of the idea), otherwise your position will get worse".