From https://en.wikipedia.org/wiki/Sophistication

> Sophistication has come to mean a few things, but its original definition was "to denature, or simplify". Today it is common as a measure of refinement

So no, it can in many cases even be the precise opposite of complexity.

It actually originally comes from "sophistry", which is an ancient greek discipline of wisdom and excellence. I would generally associate the word with a high level of complexity that has been expertly reduced and refined to an elegant quality.

Typically, different words refer to different things. Most often, words considered synonyms actually refer to slightly different things.