No, it can't be both of those things. On one analysis, it is a complement to the verb est; on the other analysis, it is the verb.
Mortuus est is always going to be analyzed as the 3rd person singular perfect passive† indicative form of morior, "has died" or just "died", not "is dead". That's what combining the participle with esse means.
Now, mortuus could just be a non-participial adjective instead. You could form a sentence that way that looked identical, ludus mortuus est, and it would mean "the game is dead", which just happens to be semantically identical with "the game has died". You can't be dead in the present without having finished dying in the past, and you can't have finished dying in the past without being dead in the present.
So we make the assumption that mortuus est is a verb, and not a predicative verb phrase, just because morior is a very common verb and this is a normal way to use it. The assumption is analogous to the assumption in English that when somebody says "time flies like an arrow", they are talking about the rapidity of the passage of time, and not exhorting the listener to apply stopwatches to flies the same way they would to an arrow.
† To the extent that morior is a passive verb. It isn't -- we call it "deponent", an active verb that takes grammatically passive forms, like the English verb to be born -- and so mortuus est is just a perfect active form.
