The first two sentences mean the same thing, and so do the last two.
(1) She's obviously the person to finish the job.
(1') She's obviously the person who should finish the job.
(2) She was the first person to finish the job.
(2') She was the first person who finished the job.
The emboldened clauses are all relative clauses, and those in (1) and (2) are infinitival relative clauses.
In general, the relative clause is not a complement but an adjunct/modifier of the antecedent. In (1) and (1'), for example, both the relative clauses are adjuncts/modifiers of the antecedent person. Clearly, they are not complements of person.
In (2), however, it seems that first licenses to finish the job. If so, is the infinitival relative clause complement of the antecedent first person? Or is it still adjunct/modifier of the antecedent just as it is in (1).
How about (2')? Does first licenses who finished the job? Is the infinitival relative clause complement or adjunct/modifier?
In short, is the relative clause always an adjunct/modifier of the antecedent? Or is it sometimes a complement of the antecedent?