The algebra needed to find the square root of a complex number is easy to imagine:$$\begin{align*}y&=\sqrt{x}\\\\\left(y_r+j\,y_i\right)^2&=x_r+j\,x_i\\\\\left(y_r^2-y_i^2\right)+j\,2\,y_r\,y_i&=x_r+j\,x_i\\\\\text{so it's just two equations and two unknowns:}\\\\y_r^2-y_i^2&=x_r\\\\2\,y_r\,y_i&=x_i\end{align*}$$ A solution will likely yield two possible answers that make sense. Each the negation of the other.
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