It's a sort of question, but not directed specifically at A; B is rhetorically playing out a question to himself.
"
Shall I go shopping?"
It might also invite input from A: "I was considering going shopping ... unless you have a better suggestion?"
ell.stackexchange.com/users/32/StoneyB on hiatus
It looks as if Person B is saying that he might go shopping, but that he is not certain.
It is common in very informal communication to append a question mark to indicate uncertainty.
In speech, this would usually be inflected as a question.
ell.stackexchange.com/users/94/ctype.h
It's an open door, not a real question.
Person B is answering person A's question by a ‘suggestion’.
If person A has something to answer to that (“I can't stand shopping, why wouldn't we go …”, which is often what he has in the back of his mind when he asks for B's plans), then the question mark is welcoming him to express it.
If person A has nothing to answer, then it's not a question for him, just a sign that person B is unsure about these plans.
At least, that's the way I use that, and I do that very often.
ell.stackexchange.com/users/57/Nikana Reklawyks
B is putting forward a possibility in front of A which he himself is not sure of and sort of expects A to give him an answer which may or may not be a definite one.
Question mark in such a scenario would also indicate that B has a certain faith in A's judgement and that he expects that his answer will not be a futile one.
ell.stackexchange.com/users/92/Mohit
I interpret this as B's question to himself, not to A, meaning "maybe I should consider going shopping?"
And it's probably not even a real question,
although the vocal pattern is the same as for a question; when written, a question mark at the end is the only way to indicate that intonation.
ell.stackexchange.com/users/39/barbara beeton
It could mean B is not sure he would go shopping.
If the question were "What should we do today?"
or "What are we going to do today?
" I would take "Go shopping?" as a suggestion for going shopping.
ell.stackexchange.com/users/95/kiamlaluno
I use a question mark all the time, to me it means all of the above comments and to emphasize the point I'm trying to make and/or
hopefully the other person will give it more thought.
ell.stackexchange.com/users/4776/Mike
Imagine the following written conversation, as recently occurred via SMS between me and a friend (edited for brevity):   A:  What are you going to do today?  B:   I don't know.  Go shopping?   What is the meaning of the question mark in the reply?  Is person B asking a question of person A?  If so, why would person B expect person A to know the answer in this case? If not, why use the question mark?
Does the question mark mean this is a question?
ell.stackexchange.com
<questions><punctuation><question-marks>
ell.stackexchange.com/questions/1042
ell.stackexchange.com/users/69/Flimzy
This is an open door, or a suggestion, not a real question.  It might also invite input in the conversation: "I was considering going shopping ... unless you have a better suggestion?". It might also be that the person is wondering if they might go shopping, but this is not certain. This is a suggestion inviting input.
This is a suggestion inviting input.