Replacing coefficients by their absolute value in continued fraction

Question

Consider a generalized continued fraction $K_{n=0}^{\infty}(a_n/b_n)$ with complex coefficients $a_n, b_n \in \mathbb{C}$. From numerical experimentation in Mathematica, I found that the distance $$ \left| \: \left| K_{n=0}^{N}(a_n/b_n) \right| - \left| K_{n=0}^{N}(|a_n| / |b_n|) \right| \: \right| $$ seems to be quite small for $N \geq 1$ and a lot of choices of $a_n, b_n$. In fact, in my experiments it will usually be less than $1$, as may be observed for instance by taking $a_n =1, b_n =1+10 i$. I also tested this for a lot of other more complicated coefficients which are polynomials in $n$ with complex coefficients (those arose naturally from a related question which is not worth reproducing here). Is there a general statement in this direction? I would be happy to learn about any special circumstances in which such a statement might hold!