I know that $\Bbb Z[i]$ and $\Bbb Z[\sqrt{-2}]$ are Unique Factorization Domains, and that $\Bbb Z[\sqrt{-6}]$ is not. I have two questions. I know that they may be difficult questions, so I only ask for books or other sources on this matter:
Is there any known result about the maximum $\alpha>0$ such that $\Bbb Z[i\alpha]$ is an UFD?
True or false?: if $x$ and $y$ are positive real numbers, $\Bbb Z[iy]$ is an UFD and $y>x$, then $\Bbb Z[ix]$ is also an UFD.
To answer your first question, if you are assuming $\alpha$ to be a real number, then there is no maximum. In particular, if $\alpha$ is transcendental then $\mathbf{Z}[i \alpha] \simeq \mathbf{Z}[x]$ and is thus a UFD. If you are restricting to the more natural situation of $\mathbf{Z}[\sqrt{-d}]$ for $d \in \mathbf{Z}_+$, then $2$ is the largest value for which $\mathbf{Z}[\sqrt{-d}]$ is a UFD. In particular, imaginary quadratic fields of class number 1 are what you seem to be looking for. Any good book on algebraic number theory will give you a good introduction into how to answer these kinds of questions.
The answer to your second question is false, as shown by the example $\mathbf{Z}[\pi i]$ and $\mathbf{Z}[\sqrt{-5}]$. The former is isomorphic to $\mathbf{Z}[x]$, hence is a UFD, but $6 = 2 * 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ shows that $\mathbf{Z}[\sqrt{-5}]$ is not a UFD.