'Pefection is attained, not when there is nothing left to add, but when there is nothing left to take away.'
Given immortality you have unlimited time to do whatever you want... and you could become head of your field, especially if it was history, after a millennia or so. You lived it, of course you know the details.
Granted, this is so long as it is the no aging but a bullet still kills you type of immortality. If you age, but do not die.. well.. eventually you're a monolithic heap of meat. Not my first choice, no.
Full invulnerability... Give that a pass thanks. It would suck more than a bit to be uable to die, and be even more unable to live. Invulnerability is Obvious to the nth degree.