Answer:
Whom
Step-by-step explanation:
"Whom" is a pronoun that refers to the object of a verb, a preposition or a clause, meaning that we use "whom" to refer to nouns that do not perform the main action but that receive that action. Therefore, when answering questions with "whom", the answer may be expressed with object pronouns, namely, me, us, you, him, her, its, them.
In the sentence "The new doctor whom I met today has a pleasing personality", "whom I met today" is an adjective clause and the object of that clause is the doctor because it receives the action of the verb "met": Whom did you meet today? I met the doctor/I met him. So, "whom" should is the correct answer because this pronoun refers to the object of a clause, and the doctor is the object of the adjectival clause.