No, an absolute phrase or clause is not part of a complex sentence. A complex sentence consists of a dependent and an independent clause; however, an absolute phrase or clause does not have a verb in it.
According to an Oxford Dictionary of English Grammar entry, an absolute clause is defined as "absolute clause A non-finite or verbless clause containing its own subject, separated from the rest of the sentence by a comma (or commas) and not introduced by a subordinator."
However, a complex sentence includes a subordinator (forming an adverb clause) or a relative pronoun (forming an adjective clause). Each of these clauses has a verb in it.
Hope that helps.