Obviously in the simplest variant it's not complex at all. I've seen amateur rocket engine nozzles shaped through applying a nail to packed clay layer.
How complex is the science behind the shape of fixed nozzles used professionally - in space-faring rockets, in military missiles, solid rocket boosters etc? How much work - comparing to difficulty of other factors - is spent on designing the nozzle for a rocket?
The general shape of fixed nozzles is pretty close to a solved problem. Per Huzel, even a conical nozzle of sufficient length and 15º half-angle gives better than 95% efficiency. Bell nozzles can exceed 99% efficiency in much shorter form.
The actual engineering design of a nozzle for a large liquid-fueled rocket is more complex; cooling the nozzle becomes a large challenge, and in regeneratively cooled engines, it interacts with the propellant pump system.
In solid rockets, regenerative cooling isn't an option, so I believe ablative cooling is more common, helped by the fact that burn times are often shorter -- typically around 2 minutes instead of 4-6 minutes for liquid first stages. Substantial engineering issues involved in interfacing the casing to vectored nozzles, but again, shape-wise, it's generally a near-optimal bell nozzle.
Either the Huzel or Sutton book will take you through the next couple of levels of complexity on the topic.
"How complex" sounds like a judgement call, but if you wish to make up your own mind, take a look at Chapter 3 "Nozzle Theory and Thermodynamic Relations" in Sutton. Although this edition of the book is somewhat infamously "watered down", this particular chapter still seems good, and should give a handle on the basic theory.
Please note that you asked about the "science", which is covered in the work I cite. If you want to talk about the "engineering" of rocket nozzles, which involves turning this theory into a workable device, that's another question entirely.
