Step 1: Understand the task.
We must rewrite the given compound sentence as a correct complex sentence without changing its meaning.
Step 2: Recall sentence types.
A complex sentence has one main clause and at least one dependent clause linked by a relative pronoun like which or that.
Step 3: Break the original.
My brother gave me a watch and I have lost it has two independent clauses joined by and, so it is compound.
Step 4: Convert one clause.
Turn one clause into a relative clause describing the watch, giving I have lost the watch which my brother gave me.
Step 5: Reject the wrong options.
Option B keeps but, staying compound; C uses since and twists the meaning into false cause; D adds an unneeded passive construction.
Step 6: Conclude.
The clean complex form is option A.
\[ \boxed{\text{A. I have lost the watch which my brother gave me.}} \]