What kind of different madness do we see in Shakespeare's major tragedies "Othello","Macbeth","Hamlet" and "King Lear"?
This is indeed a very unique question. I would say that the desire to covet is what helps to bring on the madness and descent of each character. Each character seems to covet something that they believe should be readily apparent. In their desire to bring this element of the subjective into the objective, their coveting helps to bring out their destruction in their futility in appropriating the world in accordance to their own subjectivity. Othello covets Desdemona to such an extent that he cannot fathom that something he wants to badly would actually be intended for him anyway. His coveting of her ends up representing where Iago strikes. Macbeth covets power and what he sees as being rightly his own. It is this covering that causes his breakdown in morality and the plunge into an abysmal state. Hamlet covets justice, and genuinely believes that it can be restored in the world with his fixation upon it. Achieving justice for his slain father becomes the basis for all that ends up going wrong for him and those who have the unfortunate curse of caring for him. Finally, King Lear covets a child's love and the narrow and misreading of this coveting causes him to embrace the disingenuous and shun the authentic expression of it.
An interesting question. All of these tragic heroes have a pathological tragic flaw that hinges on madness. In Othello, it is jealousy and Iago takes advantage of this to bring about Othello's downfall. Iago himself, however, is envious to the point of being pathological about it and this eventually leads to his downfall. In Macbeth, it is ambition, and his ambition destroys him when it becomes an obsession. This ambition leads to murder and guilt, and these things drive Lady Macbeth literally insane. Hamlet is morbidly depressed and because of this, he is cruel to Ophelia, and this drives her mad. King Lear becomes mad due to a combination of events that begin with his pride in insisting that his three daughters vie over expressing their love to him. He is betrayed by two of his daughters and this, along with other betrayals, drives him mad.