(1)

\begin{align*} \sqrt{\alpha^{2}} & =\sqrt{\left|\alpha\right|^{2}\sgn^{2}\left(\alpha\right)}\\ & =\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)} \end{align*}

(1)-2

\begin{align*} \sqrt{\alpha^{2}} & =e^{\frac{1}{2}\Log\left(\alpha^{2}\right)}\\ & =e^{\frac{1}{2}\left(\ln\left|\alpha^{2}\right|+i\Arg\left(\alpha^{2}\right)\right)}\\ & =e^{\ln\left|\alpha\right|+\frac{i}{2}\Arg\left(\left|\alpha\right|^{2}\sgn^{2}\left(\alpha\right)\right)}\\ & =\left|\alpha\right|e^{\frac{i}{2}\Arg\left(\sgn^{2}\left(\alpha\right)\right)}\\ & =\left|\alpha\right|e^{\frac{1}{2}\Log\left(\sgn\left(\sgn^{2}\left(\alpha\right)\right)\right)}\\ & =\left|\alpha\right|e^{\frac{1}{2}\Log\left(\sgn^{2}\left(\alpha\right)\right)}\\ & =\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)} \end{align*}

(2)

\begin{align*} \sqrt{a^{2}} & =\left|a\right|\sqrt{\sgn^{2}\left(a\right)}\\ & =\left|a\right| \end{align*}

(2)-2

\begin{align*} \sqrt{a^{2}} & =e^{\frac{1}{2}\Log\left(a^{2}\right)}\\ & =e^{\frac{1}{2}\ln\left|a^{2}\right|}\\ & =e^{\frac{1}{2}\ln\left|a\right|^{2}}\\ & =e^{\ln\left(\left|a\right|\right)}\\ & =\left|a\right| \end{align*}

(3)

\begin{align*} \sqrt{-a^{2}} & =\sqrt{\left(ai\right)^{2}}\\ & =\left|ai\right|\sqrt{\sgn^{2}\left(ai\right)}\\ & =\left|a\right|\sqrt{\sgn^{2}\left(i\right)}\\ & =\left|a\right|i \end{align*}

(3)-2

\begin{align*} \sqrt{-a^{2}} & =e^{\frac{1}{2}\Log\left(-a^{2}\right)}\\ & =e^{\frac{1}{2}\left(\ln\left|-a^{2}\right|+i\Arg\left(-a^{2}\right)\right)}\\ & =e^{\frac{1}{2}\left(\ln\left(a^{2}\right)+i\pi\right)}\\ & =ie^{\frac{1}{2}\left(\ln\left(\left|a\right|^{2}\right)\right)}\\ & =ie^{\ln\left(\left|a\right|\right)}\\ & =i\left|a\right| \end{align*}

(4)

\begin{align*} \sqrt{\pm a^{2}} & =\sqrt{e^{\frac{\pi i}{2}\left(1\mp1\right)}a^{2}}\\ & =\sqrt{\left(e^{\frac{\pi i}{4}\left(1\mp1\right)}a\right)^{2}}\\ & =\left|e^{\frac{\pi i}{4}\left(1\mp1\right)}a\right|\sqrt{\sgn^{2}\left(e^{\frac{\pi i}{4}\left(1\mp1\right)}a\right)}\\ & =\left|a\right|\sqrt{e^{\frac{\pi i}{2}\left(1\mp1\right)}}\\ & =\left|a\right|e^{\frac{\pi i}{4}\left(1\mp1\right)}\\ & =\left(\frac{1+i}{\sqrt{2}}\right)\left(\frac{1\mp i}{\sqrt{2}}\right)\left|a\right|\\ & =\frac{1}{2}\left(1+\left(1\mp1\right)i\mp\left(-1\right)\right)\left|a\right|\\ & =\frac{1}{2}\left(1\pm1+\left(1\mp1\right)i\right)\left|a\right| \end{align*}

(4)-2

\begin{align*} \sqrt{\pm a^{2}} & =e^{\frac{1}{2}\Log\left(\pm a^{2}\right)}\\ & =e^{\frac{1}{2}\left(\Log\left(a^{2}\right)+\Log\left(\pm1\right)\right)}\\ & =e^{\frac{1}{2}\left(\Log\left(a^{2}\right)+\frac{1\mp1}{2}i\pi\right)}\\ & =e^{\frac{i\pi}{4}}e^{\frac{\mp i\pi}{4}}e^{\frac{1}{2}\left(\Log\left(\left|a\right|^{2}\right)\right)}\\ & =e^{\frac{i\pi}{4}\left(1\mp1\right)}\left|a\right|\\ & =\frac{1}{2}\left(1\pm1+\left(1\mp1\right)i\right)\left|a\right| \end{align*}