Question about $rm AdS$ conformal boundary in Poincare coordinates

Disclaimer: my notation here differs from yours: My variable $z$ is your variable $1/z$. So $z=0$ for me is $z=infty$ for you and vice-versa.

If we think about the $AdS$ space as embedded on $mathbb{R}^{2,p}$ we have

$$ X^{I}X_{I}=-R^{2} $$

where the metric is the induced metric from $mathbb{R}^{2,p}$.

The boundary is defined as the limit $n_{I}X^{I}rightarrow infty$ for all possible $n_{I}$ compatible with the constraint above. This is also the place where the induced metric diverges.

Now, once we cover a patch of AdS by coordinates, and define our time coordinate, what we should look after is the intersection of boundary defined above with our patch. It is not guaranteed that our path will cover the whole space.

The boundary of the global $AdS$ have the topology of $$ frac{S^{p-1}times S^{2}}{mathbb{Z}_2} $$ Doing the universal cover in order to avoid closed time-like curves we open up $S^2$ into $mathbb{R}_t$ getting $$ S^{p-1}timesmathbb{R}_t $$ as the topology of the boundary.

The Poincaré patch is the patch of $AdS$ covered by the Poncaré coordinates:

$$ X^{+}=left(frac{1}{z}+z,x^{mu}x_{mu}right),quad X^{-}=R^{2}z,quad X^{mu}=Rz,x^{mu} $$

where $x^{0}$ is the time coordinate. Note that this does not cover the entire $AdS$ space. This coordinates have an horizon located at $z=0$, where the time coordinate never cross. You can compare with the situation of a Rindler frame for flat space.

The metric on the Poincare patch in Poincare coordinates is

$$ ds^{2}=R^{2}left(frac{dz^{2}}{z^{2}} + z^{2}dx^{2}right) $$

The part of the boundary of $AdS$ that intersect this patch can be obtained by looking at curves parameterized by the time-coordinates where the metric diverges. This happens when $zrightarrow infty$, so the intersection of the AdS boundary with the Poincaré patch is at $zrightarrowinfty$. The boundary of this patch have the topology of $$ mathbb{R}^{p-1,1} $$

An interesting thing happens when we perform a Wick rotation $x^{0}rightarrow ix^{p}$. The horizon $z=0$ closes to a point since the size of the hypersurface defined by holding $z$ fixed shrinks, i.e. become small, as $zrightarrow 0$. The same phenomena happens when we do Wick rotation on Rindler coordinates. The horizon of the Rindler coordinates also shrinks to a point.

Another important feature of the Wick rotation of Rindler/Poincaré is that they become global euclidean flat/AdS space.

If you look at the boundary of the global euclidean $AdS$ space we get the topology of $$ S^{p} $$ For the Euclidean Poincare we get the same boundary, where the point $z=0$, that was the horizon before the Wick rotation, now becoming the north pole of $S^{p}$.

Basically what happened is that under Wick rotation the boundary $mathbb{R}^{p-1,1}$ was mapped to $mathbb{R}^{p}$ and the horizon was mapped to a point at infinity of $mathbb{R}^{p}$, closing it to a $S^{p}$.