I'm in Limbo

Algebra Level 5

$\LARGE { i }^{ { i }^{ i^{.^{.^{.}}} } }$

For $i = \sqrt{-1}$ , the value of the infinitely nested exponent above is equal to $A + Bi$ for real values $A$ and $B$ .

Find the value of ${ A }^{ 2 }+{ B }^{ 2 }$ to 3 decimal places.