What kinds of algebraic integers are of degree $4$?

Among other things, you are missing $~2x~=~\sqrt{\alpha-\beta}~-~\sqrt{\beta-\alpha+\dfrac2{\sqrt{\alpha-\beta}}}~,~$ which, for $~\alpha=\sqrt[3]{\dfrac12+\dfrac{\sqrt{849}}{18}}~$ and $~\beta=4~\sqrt[3]{\dfrac2{3(9+\sqrt{849})}}~,~$ is one of the solutions of $x^4-x-1=0$.

The point being, explicit formulas for algebraic integers of degree four can be very, very complicated, much more complicated than for degree two.

$($To get another version of this number, type $x^4-x-1=0$ into Wolfram Alpha, and after it gives you a numerical solution, ask it for the "exact form".$)$

Try a nontrivial fifth root of unity, being a root of $(x^5-1)/(x-1)$. It is ultimately hopeless and not a wise use of time to try to describe all algebraic integers of degree 4.

Your description of all algebraic integers of degree 2 is incomplete also. Try $(1+\sqrt{5})/2$. There are many more examples where that came from.

I choose not to be daunted by this question, even though in some regards it is daunting, however much that has been overplayed so far. I think that a lot of these algebraic integers of degree $4$ can be boiled down to $a + b \theta + c \theta^2 + d \theta^3$, where $a, b, c, d$ are all integers, or perhaps all rational numbers satisfying a certain condition, and $\theta$ is an algebraic integer of degree $4$.

Something tells me that it is this $\theta$ that you're actually interested in, namely, your apparent ignorance of quadratic integers like $\omega$ and $\phi$ despite your earlier demonstrated acquaintance with them.

It is for these $\theta$ that things get hairy. What I have pieced together so far, mainly from your question and from comments:

• Fourth roots of integers, provided they are not perfect powers and not divisible by any fourth powers.
• Sums of two square roots of coprime squarefree integers.
• Square root of an integer plus a square root.
• Maybe the an integer plus a square root, divided by a square root or the cube of a square root?

At least we don't have to worry about Abel's impossibility theorem at this degree.