Assume that $p\equiv3\mod4$ is an odd prime and $k$ an odd number. Then $$m=(-1)^{\frac{p^k-p^{k-1}+2}{4}}$$ seems to be always the value $1$ (?). This would be interesting how one can prove this - I made some congruence reasoning but in that way I cant find a solution for the problem.
Now the core of my question : if I set $(-1)^{\frac 14}=e^{\frac{-\pi i}{4}}$ then I reach by calculation with complex numbers in exponential form a formula I would like to have $$m=(-i)(-1)^{\frac{p^k-p^{k-1}}{4}}$$
But when I do the same calculation setting $(-1)^{\frac 14}=e^{\frac{+\pi i}{4}}$ I get $$m=i(-1)^{\frac{p^k-p^{k-1}}{4}}$$ which is not the one which I am looking for.
I calculated e.g. thus $$e^{\frac{-\pi i}{4}(p^k-p^{k-1}+2)}=e^{\frac{-\pi i}{4}(p^k-p^{k-1})}e^{-\frac{\pi i}{2}}=(-i)(-1)^{\frac{p^k-p^{k-1}}{4}}$$