Is 1 or 2 correct? Or neither?
- I'm not saying I'm Batman, but answer me this; have you ever seen me and Batman in the same room?
- I'm not saying I'm Batman, but answer me this. Have you ever seen me and Batman in the same room?
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I would use either a comma or a colon or, as you've done in sentence 2, a separate sentence:
Using a semicolon would be my fourth choice.
Punctuation rules are often simply a matter of intuition, feeling, or some other type of personal style choice. I wouldn't say using the semicolon is right or wrong, just least desirable to me. Others may disagree.
I would use either a colon or an em-dash, depending on my mood. A semi-colon seems wrong to me, as the "this" in the first part of the sentence is grammatically related to the second part of the sentence. A comma also seems like a poor choice, as a larger separation is really in order.
The semicolon is just wrong: It defeats the purpose of juxtaposing the two semantic entities.
Grammar cannot say it is incorrect so long as the structure of the sentence obeys the rules of grammar. However, consider the two parts:
This introduces the context and draws attention to the main subject coming up. While so, it is semantically incomplete, so not exactly a 'sentence'. It cannot be terminated with a period; nor with a semicolon. (Once again, not in terms of grammar). Think of it as no more than Tell me.
A complete sentence. This is the subject: the point that the speaker wanted to convey, following the intro. The introducing segment literally should lead the reader here. The natural choice of punctuation for the previous part therefore, would be a full colon.
You can now see the justification for and the rationale of not using a period or semicolon between the two.