These adjectives refer to something having interconnected parts which make the whole thing difficult to follow.
Complex suggests a combination of many associated parts.
Ex – They had made such a complex system that it was difficult to submit a simple form. Commonly used with system, structure, mixture, relationship, issue
Complicated stresses elaborate relationship between the parts.
Ex – Rubik’s Cube is a complicated puzzle. Commonly used with problem, puzzle, relationship
Intricate refers to tangled parts.
Ex – The woman weaved an intricate pattern with the wool. Commonly used with design, pattern, carving, detail, melody
Convoluted is different as it means twisted and thus difficult to follow.
Ex – The plot of the movie was convoluted.
Commonly used with story, plot, scheme, argument, logic