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 

Continuing from Jamal Farokhi's answer, since $$X_{1,2}=\dfrac{1\pm\sqrt{ 1+4i\pi}}{2}$$what remains is to explicit the radical.
If $$a+i b=\sqrt{ 1+4i\pi}$$ $$a^2-b^2+2iab=1+4i\pi$$ then $a^2-b^2=1$ and $ab=2\pi$, which by the end give $$X_1=\frac{1}{2}+\frac{1}{2} \sqrt[4]{1+16 \pi ^2} \cos \left(\frac{1}{2} \tan ^{-1}(4
   \pi )\right)+\frac{1}{2} i \sqrt[4]{1+16 \pi ^2} \sin \left(\frac{1}{2} \tan ^{-1}(4
   \pi )\right)$$ $$X_2=\frac{1}{2}-\frac{1}{2} \sqrt[4]{1+16 \pi ^2} \cos \left(\frac{1}{2} \tan ^{-1}(4
   \pi )\right)-\frac{1}{2} i \sqrt[4]{1+16 \pi ^2} \sin \left(\frac{1}{2} \tan ^{-1}(4
   \pi )\right)$$
