For me, "complexity" is defined mainly by verisimilitude. The more believable and authentic a story seems, the more complex it is, imo; since the real world has endless interconnectedness and interaction on all levels of existence, even a "simple" story like a slice-of-life can be very complex indeed the more it represents a world that looks alive.
Perhaps another way of categorizing stories (simple vs. complex) should be "easy" vs. "difficult". Easy stories are ones which are straightforward to follow: slice-of-life, generic fantasies, low-key adventures, near-future technothrillers, and so on, featuring small numbers of characters, story threads, and/or multi-layered plots; while difficult stories are grand epics with tons of characters, several main arcs interwoven with subtle sub-plots for many characters/events, and have original and weird/unconvetional settings or narrative structure (or both). Difficult stories also tend to use more complex language (both in terms of words and sentence-level structure), bury a ton of details in passing mentions, and generally provoke the reader to stop and think at least once every page or so. (The really hard ones require brainwork every other paragraph or so.)
Some examples (regardless of genre or authenticity or story dramatics):
- "Easy" stories: Inheritance Cycle series, Harry Potter series, anything written by Dan Brown or Michael Crichton (both explain complex concepts really well!), Elantris by Brandon Sanderson, Fallen Dragon by Peter Hamilton, any Jack Reacher novel
- "Difficult" stories: anything written by Greg Egan (anything! though his stories are not necessarily complex, re verisimilitude), The Golden Age trilogy by John C Wright, The Dragon Never Sleeps by Glen Cook, Requiem for Homo Sapiens trilogy by David Zindell, The Forgotten Beasts of Eld, Earthsea series (Ursula le Guin has a really specific prose style which requires getting used to), Snow Crash and Diamond Age by Neal Stephenson, Malazan Book of the Fallen
Let's examine also complex vs. simple stories (aka authentic versus "two-dimensional")
- "Simple" stories: The Mastership Game by Scott Mcbain (an incredible book btw, but it is more of an allegory than a real story), Inheritance Cycle, Jack Reacher novels (they are awesome - the old ones, at least - but they are a plain power fantasy), Riftwar Saga series (a fuckton of books which are nevertheless the definition of "generic high fantasy", right up there with the Wheel of Time), anything written by Jack Vance (The only memorable thing from the Demon Princes and Planet of Adventure were how shallow the plots were, and the absolutely kickass epigraphs)
- "Complex" stories: Together We Will Go by J Michael Straczynski, Elantris, Requiem for Homo Sapiens, Heartstriker series by Rachel Aaron, The Golden Age trilogy, Fallen Dragon, Jurrasic Park and its sequel (the books are orders of magnitude superior to the movies)
As is evident, there are overlaps in these two general categories. As a rule of thumb, difficult stories tend to also be complex stories, but that's not the case always. The Dragon Never Sleeps for example is a hard book to read and follow (even though it's v. short), but in summary it's just a bog-standard space opera. On the other spectrum, Elantris by Sanderson (which is a superior version of his Mistborn series, imo) is v. easy to follow, but presents its plot and characters with such elegance and panache that makes it very much believable on a more heightened level.
And I guess I can stop here. I'm not sure if I made my definitions complete enough, but there you have it. At the very least I hope I gave you a few interesting titles to read. (I've used only book examples; if I had included other media... dude, was this post going to be much longer.)