Now, I know that different sources have different ideas for what to name terms, so I will try to explain what I see the difference as.
A Complementary Infinitive merely completes the meaning of a verb. So in the sentence "Puer clarus esse desiderat" it is very clear that "desires to be" is the full idea of the verb. So it is equally clear that this is an example of complimentary infinitive.
An Objective infinitive is an infinitive that is one that can be seen as the object of a verb, so in the sentences "the boy desires a friend" and "the boy desires to become", "friend" and "become" are filling the same role as object of the verb "want". So many times an infinitive can be parsed as both objective and complementary. In fact, the complementary infinitive is a subset of objective infinitive uses.
Now lets look at the second sentence: Puer amicum clarum esse desiderat. The boy wants his friend to become famous. in this case it is the whole phrase "his friend to become famous" that is the object of the sentence. However, "esse" still completes the idea of the main verb; "he wants his friend" would be an incomplete idea. The problem I see with this second sentence is that the structure is more different in Latin than it appears in English. In Latin, I would really prefer this as an Sentence with an indirect statement: "The boy desires
that his friend becomes famous". In this case it becomes quite apparent that the infinitive is the main verb in the Indirect Statement.
So, to summarize so far:
1) Complementary Infinitives are a type of objective infinitive use, and so are often called "compl. obj. infinitives".
2) When the accusative direct object of the main verb also seems to be the subject of an infinitive verb, you should recognize the possibility for Indirect Statement.
3) (though as of yet unexplained) Not all objective Infinitives are complementary. Some verbs are known for taking two accusative direct objects "I teach the boys math", for example has "boys" and "math" are in the accusative case. Similarly common you might see "I teach the boys to swim". "To swim" is just filling in the place of the second accusative in the double accusative structure, and is an objective infinitive. When we look so see if it is complementary though, "I teach the boys" is a perfectly fine sentence, and does not require a verb to complete the main meaning.
I hope this helps some
