How does one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-43}}{2}\right]$ is a unique factorization domain? By extending the Euclidean algorithm one can show that $\mathbb{Z}[i]$ has unique factorization.
This logic extends to show $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right]$, $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{7}\right]$ and $\mathbb{Z}\left[\frac{1 + \sqrt{-11}}{2}\right]$ have unique factorization.
However, there are principal ideal domains which are not Euclidean. How do we check that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ and $\mathbb{Z}\left[\frac{1 + \sqrt{-43}}{2}\right]$ have unique factorization?

There are two other interesting rings: $\mathbb{Z}\left[\frac{1 + \sqrt{-41}}{2}\right]$ and $\mathbb{Z}[\sqrt{-41}]$.

The first thing to keep in mind is that almost all complex quadratic integer rings are non-UFDs. Put another way, if a random $d$ is negative, then you can be almost certain that the ring of algebraic integers of $\mathbb{Q}(\sqrt{d})$ (often denoted $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$) has class number greater than $1$. If $d < -7$ is odd, then $2$ is irreducible but $d + 1 = (1 - \sqrt{d})(1 + \sqrt{d})$. For example, in $\textbf{Z}[\sqrt{-21}]$, we have $22 = 2 \times 11 = (1 - \sqrt{-21})(1 + \sqrt{-21})$, yet neither of the last two factors is divisible by $2$ or $11$.
In domains like $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})}$ and a very few cases you've probably already seen in your textbooks (e.g., Heegner numbers), things get a lot more interesting. It is true that $$20 = 2^2 \times 5 = 2^2 \left(\frac{1 - \sqrt{-19}}{2}\right) \left(\frac{1 + \sqrt{-19}}{2}\right),$$ but it turns out that $$\left(\frac{1 - \sqrt{-19}}{2}\right) \left(\frac{1 + \sqrt{-19}}{2}\right) = 5.$$ This means that $2^2 \times 5$ is an incomplete factorization of $20$ in this ring.
And as you already know, this ring is not Euclidean, so that shortcut for proving unique factorization is not available. It is worth reviewing the fact that every principal ideal domain is a unique factorization domain (Theorem $2.3$ in Peric & Vukovic). If you can prove that the ring at hand (no pun intended) is a principal ideal domain, then you have also proven that it is UFD. There is a paper where they do just that, but I can't remember at the moment how I found it (it's on the Internet, I can tell you that much).
Peric & Vukovic take a different approach: they prove that $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})}$ is an "almost Euclidean domain" (Definition $2.3$, Theorem $3.3$) and that all such domains are also UFDs (Theorem $2.2$).
