complementary IIR filters
The frequency response of the two complementary filters are $H_2(e^{j\theta}) = 1 - H_1(e^{j\theta})$, or the impulse responses $h_2[n] = \delta[n] - h_1[n]$.

For an IIR filter, $H_1(z)$ can be written as $\frac{b_0 + b_1 z^{-1} + \ldots}{a_0 + a_1 z^{-1} + \ldots}$. Then $H_2(z)$ should be something like $\frac{(a_0 - b_0) + (a_1 - b_1) z^{-1} + \ldots}{a_0 + a_1 z^{-1} + \ldots}$.

So the non-recursive coefficients for $H_2$ are now $(a_0-b_0)$, $(a_1 - b_1)$, etc.

The recursive coefficients are the same for both filters.