Asymptotic analysis of a real-valued function involving complex numbers
Only the root(s) with largest magnitude matters (for generic initial conditions), and the asymptotic rate of growth is exponential in this magnitude (assuming no repeated roots). In your case this root is real, and it's a simple exercise to show that the others don't affect the asymptotic rate of growth.

A more complicated case is recurrences such as $a_n = a_{n-2}$. In this case there are two roots of largest magnitude, both complex. This corresponds to the periodic nature of the recurrence. Nevertheless, for generic initial conditions, the growth rate is $\Theta(1)$, agreeing with the first sentence above.