Does Lagrange interpolation work for complex numbers?

Question

I am attempting a question that regarding Polynomial and remainder theorem with quadratic divisors ( $x^2 + 1 $) and ($x^2 + 2$) specifically. The resulting remainder is supposed to be a degree 3 polynomial. However, since the quadratic divisors have imaginary roots, would it be possible to use Lagrange interpolation for complex numbers? e.g. R(i) = 3i+2, R(-i) = -3i+2 and (with tedious manipulation) obtain the answer. Is there a better method to compute, and is it Mathematically correct to use Lagrange interpolation to determine the polynomial with complex roots?