There aren't many aspects of modern psychology and philosophy that weren't first seen in Hamlet. In fact, just how many fields of enquiry did this play open? In terms of psychology, there are scenes that deal with depression, manic-depression, Freudian theory, the stages from Elisabeth Kubler-Ross' On Death and Dying... there's even a moment that seems to foreshadow the Rorschach test (with clouds substituting for ink at the end of 3.2). As for philosophy, and the literature based on it, I'm going to quote the ending of the essay on Hamlet from Northrop Frye on Shakespeare (1986), which is one of my favourite books:

"During the nineteenth century, and through much of the early twentieth, Hamlet was regarded as Shakespeare's central and most significant play, because it dramatized a central preoccupation of the age of Romanticism: the conflict of consciousness and action, the sense of consciousness as a withdrawal from action which could make for futility, and yet was all that could prevent action from becoming totally mindless. No other play has explored the paradoxes of action and thinking about action so deeply, but because it did explore them, literature ever since has been immeasurably deepened and made bolder. Perhaps, if we had not had Hamlet, we might not have had the romantic movement at all, or the works of Dostoyevsky and Nietzsche and Kierkegaard that follow it, and recast the Hamlet situation in ways that come progressively nearer to us. Nearer to us in cultural conditions, that is, not in imaginative impact: there, Shakespeare will always be first."