Given $\{a, b, c, d, e\} \in \textbf Z$, it seems to me that any number of the form $$a + b i + c \sqrt{-2} + d \sqrt 2 + e (\sqrt 2 + i)$$ should be an algebraic integer in $\textbf Q(\sqrt 2 + i)$. But I doubt this is a complete characterization, there might be cases $\{a, b, c, d, e\}$ can be drawn from $\textbf Q\setminus\textbf Z$, or maybe I have overlooked algebraic integers by which to multiply the integers, maybe both.
The first thing I did was look at some positive integer powers of $\sqrt 2 + i$, like $(\sqrt 2 + i)^5$ (I also looked at negative integer exponents but am not sure what, if anything, I should make of them).
Then, figuring that $1 + \sqrt 2$ must be a unit in this domain, I verified that $$\frac{\sqrt 2 + i}{1 + \sqrt 2} = 2 - i - \sqrt 2 + \sqrt{-2},$$ $$\frac{\sqrt 2}{1 + \sqrt 2} = 2 - \sqrt 2,$$ $$\frac{i}{1 + \sqrt 2} = -i + \sqrt{-2}$$ and $$\frac{\sqrt{-2}}{1 + \sqrt 2} = 2i - \sqrt{-2}.$$
I acknowledge that what I have tried may be insufficient, or altogether on the wrong track. What have I overlooked, or what should I have been doing, to find the algebraic integers of this domain?