Certainly that famous first line of Hamlet's fourth soliloquy is an existential question: To exist or to not exist. Hamlet wonders if it is not to suffer all the ills of living--"the slings and arrows of outrageous fortune"--or end this suffering--"or to take arms against a sea of troubles,/And by opposing, end them. Could one sleep if one opposes the troubles? What dreams could one have in the afterlife?
Hamlet concludes that because one does not know what follows death, one continues to live. It is the "dread of something after death" that keeps men from ending their suffering.
The thinking of what is to come gives pause to one's resolution of "not to be." And, "Thus, conscience doth make cowards of us all."
Hamlet's puzzlement comes from his debate of which is better: to suffer in life, or to end one's suffering and risk the consequences that can be worse in the hereafter. His conscience--Hamlet knows it is a sin to commit suicide--stalls his hand, and again his dilatory nature wins out.
The major questions that Hamlet brings up are about mortality and whether or not it is worth the trouble of dealing with difficulty, etc., "the slings and arrows of outrageous fortune" or if it would be easier just to kill yourself.
The fascinating part, to me, is the idea of this "undiscovered country," and the fact that death is likened to sleep but a sleep in which you might dream. This blend of reality and questions about the idea of dreaming being the reality of death, etc., is very interesting.
So is it in fact a simple question of whether it is worth it or not, or is it also important to examine what death might actually be before jumping into it with both feet?
Hamlet's life is in a really bad place. His father has been murdered and his mother quickly re-married the murderer, His girlfriend has dumped him. His friends have proved disloyal. He should be king but his Dad's killer is king instead. And all the ideaologaical values of beauty, justice and truth that he cherished in his youth are now empty and meaningless.
Life has given Hamlet a massive wake-up slap from his pampered and privileged childhood. And he is saying to himself, "I hate this, why don't I simply end it?" Why should I live a grey and unhappy life full of troubles and pain? Why don't I just kill myself.
Hamlet's reason are not complicated or particularly deep. But they are beautifully expressed. He lists all the bad things life has and all the reasons death would be pleasant. In the end he concludes that the only reason he won't kill himself is becuase God says it is forbidden and life after death might be even worse than life. To leave this world is only a good idea if the next is better or nothing at all. But it might be worse.
Thus conscience does make cowards of us all,
And thus the native hue of resolution
Is sicklied o'er with the pale cast of thought,
And enterprises of great pitch and moment
With this regard their currents turn awry,
And lose the name of action.
The only reason he doesn't kill himself is too much thinking. If he were brave and purposeful, he would do it.
(but, of course, he is wrong; a bad life is better than death because a bad life can get better. Whereas death is unchanging.)