I'm not sure if it's been established what it really means for a doctor to regenerate, whether they just change or if in fact it's an actual death for them and it's just that their memories and knowledge get passed onto a new being.

If the latter I would understand if the doctor actually goes through an existential crisis at each regeneration. The doctor might live on, but for them personally it's the absolute end each time.

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I do think that the 10th Doctor's end was a bit too melodramatic though. It was a really stupid death for one thing and it felt as though they were giving 10 a special treatment. I also find the last line jarring given how as far as I can tell David Tennant chose to leave the role.