#### Answer

$a=\pm i$

#### Work Step by Step

Since $x^2=a$ implies $x=\pm\sqrt{a}$ (the Square Root Principle), then the solutions to the given equation, $
a^2+1=0
,$ are
\begin{array}{l}\require{cancel}
a^2=-1
\\\\
a=\pm\sqrt{-1}
\\\\
a=\pm i
.\end{array}