Complex adaptive systems (CAS) are coevolving, multiplex networks. The study of CAS is inherently interdisciplinary and leans heavily on algorithmic methods (as opposed to purely analytical ones). This is because CAS are highly variable/nonlinear (~whole > sum) and the coevolution of multidimensional interactions with the elements results in a vast phase space.
CAS can also be thought of as the study of generalized interactions between generalized matter. For example, the "matter" can refer to people, nations, agents, simulacra, cells, etc. which can interact via "forces" mediated through the exchange of signals such as information, glances, pheromones, hugs, etc. They are outofequilibrium processes which have selective interactions based on these signals. Due to the presence of selection processes which replicate and refine components/modules, CAS often operate under evolutionary dynamics, continuously change their boundary conditions, are selforganized critical/operate at the edge of chaos, are path dependent, and have memory. They are a superposition of many, plastic interaction networks of varying strengths.
The particular nature of a CAS' elements and its interaction types define what kind of 'universe' it generates and implicate endless potential trajectories of abstract tilings.
Clear signs something is a CAS:

it is adaptive and robust at the same time

nonlinear dynamics and chaos (i.e. too hard to describe analytically)

aggregation, self organization, hierarchies, emergence, metaagents

discrete actions/optimization conditioned on signals

internal models

modules or niches

networks, loops, and flows

diversity

phase changes, power laws, and selforganized criticality

collapses and booms

high assembly index/complextropy
Examples:

LLMs

Simulacra

Ecosystems and biomes

Biological, cultural, technological... evolution

Human brains (and, of course, all our organs and organ systems)

Nations, cultures, companies etc.

Ant colonies

Genetic regulatory networks

Intergalactic alien societies

Future superintelligences
Tools for studying/engineering CAS:

Thinking about it on your own

Nonlinear dynamics

Probability theory, stochastic processes

Scaling and power laws

Computation, genetic algorithms, complexity theory, information theory, statistical mechanics

Dynamics/science of: networks, evolutionary processes, economics, and machine learning algorithms

Data science, agent based modeling, computational physics, generative deep learning