Well, I've detailed what I've thought of as more complex portrayals of the Klingons, particularly during TNG and DS9 earlier in this thread: differing internal politics, different classes, differing ideas of honor and how to achieve it. Obviously, we don't need to agree.I always liked Errand of Mercy because they have to have an external power say they're being stupid. Kor just wanted to fight as a proof of his strength and Kirk is convinced they're just wrong.
Not sure what complexity is there so I'm with how Klingons have variations of simplicity over complexity.
I will agree that the Organian intervention was fine given the nature of television at the time. I wouldn't be so enamored with it in a modern series.