Is "depth" simply a code word for for "complicated?" I don't think so.
I tend to use "complex" as a synonym for "deep" since in many cases depth arises from complexity.
However, that doesn't have to be the case, and it isn't always so. Sometimes depth arises from a few simple rules that interact in interesting ways. John Conway's Game of Life, for example, has only four rules and a couple of metarules, but when you see it in action you'd swear there must be a huge -- and complex -- set of rules controlling the behavior of the visible elements.
In reality, there is complexity, but it's hidden -- it's in the rules that allow for interaction of the rules, and in particular in the metarule that says information about the previous state is used to generate the next state. The same could be said of the original SimCity. There aren't that many rules, but their interactions create the appearance of depth. Even so, the rules are simple.
That acknowledged, I think I'm safe in saying it's still the case that most systems that seem to have an amount of depth we might call "interesting" are based on complex systems that interact (and generate new states from information about the previous state) to produce even more interesting complexity.
Still, the point is well taken. I agree; a perception that the game is "deep" is what some people -- primarily, I would say, those interested in Knowledge power -- are looking for in their games. "Complicated" is usually more descriptive of the game's interface than the gameplay itself, so "deep" really is probably the better term for gameplay (regardless of the interface) that reveals more of itself as you explore it.
That's the kind of gameplay I hope to see in a polished MMORPG someday.