P.S. In a faintly related vein, if we define i^z := exp(pi*z*i/2), then iterating this function on the starting value of z = i approaches a limit L of approx. 0.6528812343931018 + i 0.3675743023883531 (or so says my C program).

and for further information see entries A077589 and A077590 in the OEIS! Best regards Neil