Solution
= Solution
Write a <square root> as $u+iv$, with $u,v$ real. Equating the <real part> and <imaginary part> of its square gives $u^2-v^2=-1$ and $2uv=1$. Taking <complex modulus> also gives $u^2+v^2=\sqrt2$. Therefore
$$
u^2=\frac{\sqrt2-1}{2},\qquad v^2=\frac{\sqrt2+1}{2}.
$$
Since $uv>0$, the signs agree. The two roots are
$$
\boxed{\pm\left(\sqrt{\frac{\sqrt2-1}{2}}+i\sqrt{\frac{\sqrt2+1}{2}}\right).}
$$
The $\pm$ changes both signs together, rather than independently.