Optimization for inequalities in $\mathbb{C}$

Question

I want to check when the expression

$$ \sqrt{\sqrt{\lambda^{2}\left(2E+\lambda^{2}\right)}+E+\lambda^{2}} $$

is real, when it is purely imaginary and when it is complex (with imaginary part not zero).

Also I would like to know the sign of the imaginary part in the case it is non-zero. To check the last part, after some steps, I am using Reduce for inequalities like $\mathrm{Im}\left(\sqrt{\lambda^{2}\left(2E+\lambda^{2}\right)}\right)>0 $ but this is taking a lot of time on Mathematica. Is there a way to get the kind of results that I want more quickly?

The variables are real and the code I ran is

  Reduce[ Im[ Sqrt[ e + λ^2  -Sqrt[ λ^2 (2 e + λ^2)]]] > 0, {e, λ}]

Answer 1

Let's see what are the regions of interest:

n = 4;
Partition[
   RegionPlot[#[[2]][#[[1]][Sqrt[e + x^2 + Sqrt[(x^2) (2*e + x^2)]]], 0], 
             {x, -n, n}, {e, -n, n}, PlotLabel -> #, 
             AxesLabel -> Automatic, LabelStyle -> Medium] & /@ 
                                                 Tuples@{{Re, Im}, {Less, Greater}},
   2] // Grid // Framed

So in principle we should only focus on the sign of the imaginary part. Like this:

Reduce[Im[Sqrt[e + x^2 + Sqrt[(x^2) (2*e + x^2)]]] == 0, e, Reals]
(*
 e >= -(x^2/2)
*)

and that's the information you were after.