Gaussian Integers and Finite Fields

Question

I have been doing a little reading in elementary number theory, in particular I was reading about the Gaussian Integer, $\mathbb{Z}[i]$. My question regards the algebraic structure of $$\frac{\mathbb{Z}[i]}{p \mathbb{Z}[i]} $$ where $p$ is a prime in both $\mathbb{Z}$ and $\mathbb{Z}[i]$. In particular I am asking if there are any analogies I can draw from the finite field $$\frac{\mathbb{Z}}{p \mathbb{Z}} $$ where $p$ is a prime in $\mathbb{Z}.$ It is well known that $$\frac{\mathbb{Z}}{p \mathbb{Z}} $$ is a finite field iff $p$ is prime. Can anything be said $\frac{\mathbb{Z}[i]}{p \mathbb{Z}[i]}$? My hunch is that there are also finite fields. If they are, is there a way I can compute them?