Question: Square root and branch cut

For sqrt(-1), Maple returns I. Why not -I? I understand why in general Maple does not, and should not, return both signs, because sqrt is defined with a branch cut - specifically out along the negative real axis:

FunctionAdvisor(branch_cuts,sqrt(s));

But as +I and -I lie symmetrically around the branch cut, I do not understand why +I should be chosen in favor of -I. Neither the square of +I or of -I crosses the branch cut, which is,  I guess, the standard way to select a unique value, although both squares end up on the branch cut itself - the latter fact leading to the following more general consideration:

I do not understand why the square root of any negative real number (the above being just a specific case) should at all be assigned any meaning when lying as they do on the branch cut itself. I think it would be more sensible if Maple raised an error, telling you that the branch cut needs to be changed/moved if any value is to be assigned. Which leads me to the following question:

Can the branch cut of the logarithm, and thus of sqrt as well, be changed/moved? I would like it to lie out along the negative imaginary axis.

Update I: Concerning -1 lying on the branch cut itself, there is no issue, see my 'ups' in the reply to the answer by John May. But redefining the branch cut is still relevant.

Update II: Perhaps the issue raised is not all that trivial, after all. At least, it is "a subject of papers and debate", as Alejandro Jakubi formulated it in an email to me, pointing me to the article 'Function evaluation on branch cuts', by Rich and Jeffrey.