Are all roots of polynomials complex numbers? I'm sure this is a terrible question but I'd like to make sure. Given a polynomial, will its roots always be complex numbers? What else could they even be?

Usually one considers polynomials within a (number) system. Then you ask yourself whether there are roots within that system. 
The prototypical example is with the usual number systems, where one enlarges the system (integers, rationals, reals, complex) until within the complex numbers we get the fundamental theorem of algebra: the every polynomial with complex coefficients has a complex root. 
But, in more generality, one can consider polynomials with whatever objects can be added and multiplied: this applies to the coefficients and also to the "variable": and they don't have to be the same. As an example, since square matrices can be added and multiplied, and also multiplied by scalars, one can consider polynomials with complex coefficients but where the variable is a square matrix: things like
$$
p(A)=3A^3-2A^2-A.
$$
