Consider $\Omega=\{z\in\mathbb{C}:|z|<1\}$ and let $\mathcal{F}=\{f(z)=z+n:n\in\mathbb{N}\}$. We have that $\mathcal{F}$ is a family of holomorphic functions defined over $\Omega$ whose range omits infinite points of $\mathbb{C}$; hence by the Fundamental Normality Test we can say that $\mathcal{F}$ is a normal family, but this family is not uniformly bounded under compact sets of $\Omega$.
I am sure it has to be something obviuos but I have been thinking for a while and I do not find any clue of why my reasoning is wrong.
What am I missing here? Any help would be appreciated.
This family is certainly not normal with respect to $(\Omega,\mathbb{C})$, as it is not locally uniformly bounded. Indeed, take any compact subset $K$ of $\Omega$ that contains the origin, and suppose the family $\{f_n\}$ were uniformly bounded by some $M>0$. Consider the image of $0$ under the family. If we choose $N$ so that $N>M,$ then $|f_N(0)|=N>M.$
Read your version of the fundamental normality test again carefully and make sure that it applies exactly. Does it apply to $(\Omega,\mathbb{C})$ or to $(\Omega,{\hat{\mathbb{C}}})?$ Is the question about normality with respect to $(\Omega,\mathbb{C})$ or to $(\Omega,{\hat{\mathbb{C}}})?$
Recall that normality can be defined on the Riemann sphere, in which case the above work is not sufficient. In this scenario, we either need that a subsequence converges uniformly on each compact subset as a sequence into $\mathbb{C}$ (we can rule this out) or that it converges to $\infty$ uniformly on compact subsets.
This part is true. Fix any compact $K\subset\Omega.$ Note that if $z\in K,$ then $$|f_n(z)|=|n+z|\geq n-|z|\geq n-1\rightarrow\infty$$ uniformly in $n$ as $n\rightarrow\infty$. This is because the FNT that you've seen was likely applied to $\hat{\mathbb{C}}.$