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.
* 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?)
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