A "complex" story is only a good thing provided it's told in an intelligent and clever manner - if the writer manages to accomplish this, then it usually leads to a higher level of engagement and interest from the audience, since they'll constantly be evaluating what's going on.
As for "depth", that helps to make the story worthwhile. Telling a great story is one thing, but having a message or a theme(s) will have a much longer lasting impact on the audience - for one, it'll give them something to think about after they've finished reading/watching/playing through the story, as well as engaging them on an intellectual level while they're actually experiencing the story.
The character part of your question is quite simple, really - well-rounded characters are far more interesting, so once again you're generating interest in your story through these characters, as well as increasing the likelihood that your audience will relate to one of these particular characters. This, also, increases the level of engagement the audience has with the story.