Is :$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$ irrational or transcendental or real number? Is there someone who can show me if :$$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$$ is irrational or real or transcendental number ?

Thank you for any help
$$X=\sqrt{ i\pi+X}$$
$$X^{2}=i\pi+X$$
$$X^{2}-x-i\pi=0$$
$$X_{1}=\dfrac{1+\sqrt{ 1+4i\pi}}{2}$$
$$X_{2}=\dfrac{1-\sqrt{ 1+4i\pi}}{2}$$
$X_{1}$ and $X_{2}$ are not real.
Let’s call $a=\sqrt{i\pi+\sqrt{i\pi\dots}}$
\begin{align}
a^2 &amp;= i\pi+a \\
a^2-a &amp;= i\pi \\
(a^2-a)^2 &amp;= -\pi^2 \\
(a^2-a)(a^2-a) &amp;= -\pi^2
\end{align}
Solve for $a$.
Hope that helps!