Literally 'undying bird' — the translated Western phoenix. It reads as a translation.

What a Japanese person actually thinks
Reads like a term from a fantasy novel or a sports headline about a comeback. Perfectly clear, not at all Japanese in feel.

Meaning

Literally: not + die + bird
Actually: The Western phoenix, rendered into Japanese. Correct and understood, but visibly an imported concept.

Scores

Naturalness●●●●●

A real word, used mostly for the Greek phoenix and for metaphorical comebacks.

Coolness●●●●●

Three characters, decent balance. The 不死 ('undying') opening is a bit blunt.

Watch out

lowIf you want a *Japanese* mythological bird, this isn't it — 鳳凰 is. If you want the rise-from-ashes story specifically, this is the right one.

Get this instead

鳳凰 hōō
the East Asian phoenix

The genuinely Japanese bird, if the culture matters more than the exact myth.
再起 saiki
to rise again after a fall

If the comeback is the actual meaning, this says it directly and with real dignity.

On skin

不死鳥

Orientation: vertical

Mincho.