About gaussian integers and orders.

Question

I noticed $(1+i)^{16}= 256$ so $(1+i)^{16} - 1$ is a multiple of $17$. So $(1+i)^{16} - 1$ is a multiple of $(1+4i)$ or $(1-4i)$. $(1+i)^{|1+4i|}$ is congruent to $1$ or $i$ mod $(1+4i)$. I think . Any gaussian integer $W$ that is not a multiple of $(1+4i)$ will have a least integer exponent $h$ that 'makes' it congruent to $1$ or $i$. $W^h$ is congruent to $1$ or $i$ mod $(1+4i)$, and $h$ divides the norm of $(1+4i)$. So $h$ is 'like' the order of $W$ mod $(1+4i)$. Is all this feasible?