Question

What is the function of the complement shown in parentheses in this sentence "We named our (Australian shepherd) Allie" ?

Answer 1

"Australian Shepherd" is the Direct Object in the sentence.

Explanation:

Subject of sentence: We
Verb : named
Direct Object : shepherd
Modifiers of Direct Object: our Australian
Object Complement: Allie (Proper Noun)

[I would consider just "shepherd" to be the D.O., because there are different kinds of shepherd dogs. What kind of shepherd dog? A German shepherd; an Australian shepherd.]

The Original Answer is correct, but perhaps somewhat terse:
subject + Factitive Verb + Object + Objective Complement/Noun Complement/Factitive Object

Original Explanation:
We named OUR AUSTRALIAN SHEPHERD Allie.
OUR AUSTRALIAN SHEPHERD == Object (not complement)