2005 AMC 12B Problem 22

A sequence of complex numbers z_{0}, z_{1}, z_{2}, ... is defined by the rule

z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},

where \overline {z_{n}} is the complex conjugate of z_{n} and i^{2}=-1. Suppose that |z_{0}|=1 and z_{2005}=1. How many possible values are there for z_{0}?

\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 2005 \qquad \textbf{(E)}\ 2^{2005}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.