* The bird came alive and flew away.

(What is the part of speech of 'alive'? Is it a subjective complement? 'Came' is a complete intransitive verb, right? Can a complete intransitive complement take a complement?)

In this sentence, "alive" is an adjective, and yes, it's a subject complement. The verb "came" is definitely intransitive, but in this sentence it seems to be acting like a linking verb in a way.

http://grammar.ccc.commnet.edu/grammar/verbs.htm#linking
Take a look at the second set of sentences here.

In the sentence, "The bird came alive and flew away," the word "alive" is an adjective modifying the subject "bird." Specifically, it describes the state or condition of the bird. In this context, "alive" does not function as a subjective complement.

As for the verb "came," it is a complete intransitive verb because it can express a complete thought without requiring an object. In this sentence, "came" indicates the movement or change of state of the bird.

Regarding your question about a complete intransitive verb taking a complement, the answer is no. A complete intransitive verb does not have the ability to take a direct object or a complement. It only requires a subject to form a grammatically correct sentence.

To reiterate:
- "Alive" functions as an adjective modifying the subject "bird."
- "Came" is a complete intransitive verb expressing the bird's movement or change of state.
- A complete intransitive verb does not take a complement.