What is the definition of a "complementary filter"?
Usually, a complementary filter (like a complementary function) complements another filter.  The two filters that are complementary to each other add to one.  Or, at least, add to an all-pass filter (which is what Linkwitz-Riley crossovers do.

so either

$$ H(f) + G(f) = 1 $$

or 

$$ H(f) + G(f) = A(f) $$

for $H(f)$ and $G(f)$ being complements of each other and $|A(f)|=1$ is an APF.