“What about…?” and “How about…?” are idiomatic expressions that are meant to ask a question. It is true that they are not grammatically complete sentences in that they do not have a verb. However, idioms are unique unto themselves, and these two indicate questions and must stand alone with a question mark.
If there is a question after this expression, then two question marks are needed.
…What about John? Was he there?
…How about the date in question? Do you agree it is correct?
These are separate questions as there would be separate answers.
…What about John?
ANSWER: I don’t know anything about him
OR: I don’t know him.
OR He was working at the time.
…Was he there?
ANSWER: The only answer could be a positive or negative response.
So since the answers are totally different, there have to be two questions and two question marks.
Here is the definition from Merriam-Webster:
Definition of what about
1 : does that include (someone) : how about (someone) “We’re all going to the beach.” “What about Kenny?”
2 : how does that affect (someone or something) : what should be done about (someone or something) “I need to leave—something has come up.” “What about the meeting?” “We’ll have to reschedule it.” What about the people who can’t afford health insurance?
3 —used to make a suggestion about what could be done We’ll need to talk about this again. What about (meeting) next week?There’s still time. What about playing another game?
4 —used to ask someone to tell one something in response to the thing that one has just said I like skiing and hiking. What about you? What sports do you like? Everyone else is coming. What about you?