I have thought about complexity only a little, mostly with regard to two particular aspects: (1) defining complexity in a simple toy model and (2) defining the difference between complexity and complicatedness.
1. Wolfram classified elementary cellular automata (ECA) in four categories: simple, periodic, chaotic, and complex (see, e.g., page 2 at https://arxiv.org/pdf/1304.1242.pdf). Langton suggested a quantity lambda calculated from the automata rules that would be a measure of the closeness to the “edge of chaos.” Langton’s lambda works only partially for distinguishing complex ECA. Sun suggested a slightly different quantity mu (unpublished) that works a little better than Langton’s lambda. A certain combination of Langton’s lambda and Sun’s mu works better than either separately, but it is still imperfect. I have also explored a quantity based on “Garden of Eden” states for the ECA rules. My intuition is that if we cannot produce a nontrivial definition of complexity that reliably selects complex (and only complex) cellular automata, then we don’t really know what complexity is. Of course, if we could do that, it would not necessarily mean that we would know what complexity is in more realistic systems, but at least we might have a solid beginning to understand.
2. There are some who suggest that modern civilization is approaching a collapse because it is becoming too complex. I think this is a misunderstanding of what I would call complexity. Instead, I think modern civilization is becoming too complicated and insufficiently complex. I think the complicatedness and complexity are different. I have not yet reached acceptable definitions of these two concepts. Ultimately, given numerical measures of the complexity and complicatedness of a system, I would say that the fragility or resilience of the system is measured by the logarithm of the ratio of the complexity to the complicatedness of the system. If the ratio is greater than 1, then the sign of the logarithm is positive, the system is resilient, and the numerical value of the logarithm is a measure of how resilient the system is. If the ratio is less than 1, then the system is fragile, and the absolute value of the logarithm is a measure of how fragile the system is. One feature that I think is important is the cycles in a network diagram of a system. I want to find a way to represent cycles as fundamental objects in a representation of a system. Then what we have usually regarded as fundamental objects in a system (e.g., species in an ecosystem food web) would become composite objects comprising the cycles in which they participate (with each cycle being a fundamental object in the representation).
Perhaps early biochemical evolution had a long “language” history before the written RNA/DNA language was invented. Humans had a long language history before written human language was invented. In both cases, the invention of written language allowed a significant increase in the rate of evolution (biochemical evolution in one case and cultural evolution in the other). This focuses my attention on written language as an important emergent feature.