2008 AMC 12B Problem 19

A function f is defined by f(z) = (4 + i) z^2 + \alpha z + \gamma for all complex numbers z, where \alpha and \gamma are complex numbers and i^2 = - 1. Suppose that f(1) and f(i) are both real. What is the smallest possible value of | \alpha | + |\gamma |?

\textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad

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