Your question is "What am I?" I'm guessing the answer is
"to be or not to be, that is the question".
I am the question that no unerring answer knows
The thought of which does ofttimes give my asker pause
Both choices are open; both have pros and cons. The question is reflective, hence "give my asker pause".
"... Must give us pause."
When resolution wavers, he'll reflect
On which solution heretofore considered is correct
Regardless of which is chosen (to be / not to be), the decision is questioned when the resolution to go through with it wavers.
"... Whether 'tis Nobler in the mind to suffer The Slings and Arrows of outrageous Fortune, Or to take Arms against a Sea of troubles, And by opposing end them: ..."
Of heartache, of fortune, of death and dreams
In this oration, life's greatest riddles are the themes
In Shakespeare's original, Hamlet's question is framed against his father's murder (heartache). It's been a long time since I studied Hamlet, so a lot of this is rusty, but fortune may be a reference to his uncle assuming the throne. Death and dreams are part of the line "to sleep, perchance to dream".
On fortune: "... Slings and Arrows of outrageous Fortune, ...".
On heartache, death and dreams: "... The Heart-ache, and the thousand Natural shocks That Flesh is heir to? 'Tis a consummation Devoutly to be wished. To die, to sleep, To sleep, perchance to Dream; ..."
You hear me spoken time and time again
For centuries my lasting magic still remains
An allusion to the line "to be or not to be" being used in other contexts, or perhaps being quoted repeatedly due to the play being popular.
Fit for a prince, uttered in earnest, sometimes in jest
Of all my kind I surely am one of the best
The character voicing the line was Prince Hamlet. Earnest vs jest may be an allusion to how the line is used by other people in other contexts. Also Shakespeare (Hamlet's playwright) is "widely regarded as the greatest writer in the English language and the world's pre-eminent dramatist", and among his plays, the line "to be or not to be" is one of the best known.
Now there's a bounty on my head
A bit more work, and I'll be dead
"For in that sleep of death, what dreams may come, When we have shuffled off this mortal coil,"
Come on Puzzling, you can win
If you remember all your sins
"... The fair Ophelia? Nymph, in thy Orisons Be all my sins remembered."
I puzzle the will, confound the sense
'Tis time to end all the suspense
"... The undiscovered Country, from whose bourn No Traveller returns, Puzzles the will, And makes us rather bear those ills we have, ..."
The First Quarto version uses "... puzzles the brain, and doth confound the sense, ...".
About 'Tis: "... Whether 'tis Nobler in the mind ...".
References taken from
wikipedia, First Folio text except for one First Quarto quote.