Is it two or three ellipsis when using it with a question mark?

Solution 1:

From Wikipedia:

The most common form of an ellipsis is a row of three periods or full stops (. . .) or a precomposed triple-dot glyph (…). The usage of the em dash (—) can overlap the usage of the ellipsis, especially in dialogue. Style guides often have their own rules governing the use of ellipses. For example, The Chicago Manual of Style recommends that an ellipsis be formed by typing three periods, each with a space on both sides. [emphasis added]

There is no universally accepted standard for the form or use of the ellipsis. See Wikipedia, a more authoritative source, or any of a number of style manuals for additional information on the use of spaces, the role of periods, etc.

I agree with @Dan that the question mark is in addition to the ellipsis, which is its own mark of punctuation. Your second example with three dots represents common usage.