Are either of these a complex sentence? "Released from prison, the woman went to New Start for directions." "Anyone who loses a school id card must pay for a new one."

Both sentences are complex sentences.

Explanation:

A complex sentence is one that has an independent (main) clause and one or more subordinate clauses.

The independent clause can stand alone and still make sense, while the subordinate cannot stand alone as it is dependent upon the main clause to make sense.

First sentence: "Released from prison, the woman went to New Start for directions."

Here, the first part of the sentence needs context cannot stand alone, because "Released from prison" (subordinate clause) immediately begs the question of "who was released from prison?". The second part "the woman went to New Start for direction" (independent clause) can stand alone and still be understood, and it provides the context needed to understand the first part.

Similarly in the second sentence: "Anyone who loses a school id card must pay for a new one." "Must pay for a new one" (subordinate clause) immediately begs the questions of "who must pay and for what?", and therefore can not stand alone as it needs context to be understood. This context is provided by the first part "anyone who loses a school ID card." (independent clause).

The first sentence " Released from prison, the woman went to New Start for directions"  is a complex sentence

Explanation:

A complex sentence is sentence that it made up of a main clause and a subordinate clause. the main clause in a complex sentence can can regarded a simple sentence because it has an independent meaning without the subordinate clause while the subordinate clause cannot be considered a simple sentence because they literally have no meaning and can't be understood without the main clause.

The main clauses in the sentences are : " The woman went to New start for directions "

The subordinate clauses are :  " Released from prison "