Normally, branches of the logarithm are generalized in the form of $$ln_k(z)=ln(z)+2 \pi i k.$$
However, is there a such a generalization that can be made not just for integer values, but for any value? It seems there should be, and I don't quite see why they don't simply extend it to any number with such a simple looking formula.
All you would really be doing is shifting the phase along the logarithm's Riemann surface. So, take for instance $ln(z)+ \pi i k$. I don't think it would be completely accurate to express it this way, but so you can see what I am saying, can this be expressed as $ln_{1/2}(z)$?
No. The reason behind the $2\pi ik$ lies in the fact the, given $z,w\in\mathbb C$,$$e^z=e^w\iff(\exists k\in\mathbb{Z}):z=w+2\pi ik.$$Thherefore, the only branches that can exist are those that you described.