Here is an example. Sorry it's so complicated. (There's probably a simpler one, but my mind works in complicated ways, it seems.)
Consider a Siegel modular threefold $U$, i.e., a Shimura
variety for the general symplectic group $GSp(4)$ with some level $n\geq 3$.
(So that $U$ is smooth and quasi-projective over $\mathbb{Q}$.) Let $X$ be
the Baily-Borel-Satake (aka minimal) compactification of $U$. Then $X$ has
a stratification $X=U\cup X_1\cup X_2$, where $X_1$ is a finite disjoint
union of modular curves and $X_2$ is a finite disjoint union of points
(spectra of abelian extensions of $\mathbb{Q}$). Let's denote the inclusion
of $U$ in $X$ by $j$, and define the intersection complex on $X$ by
$IC=(j_{!*}\mathbb{Q}_\ell[2])[-2]$ (I like my intersection complexes to be
concentrated in nonnegative degree). Then I say that the cohomology of the stalk of
$IC$ at a point of $X_2$ is not pure (or pointwise pure, as you put it).
Before I do the messy calculation, let me give a heuristic reason. If we
allowed ourselves to restrict everything to $Spec(\mathbb{C})$ and use $\mathbb{C}$-coefficients (which we can't
because we want to look at weights of Frobenius), then we could use the
calculation of the stalks of the intersection cohomology complex by
Goresky, Harder, MacPherson and Nair in their article Local intersection
cohomology of Baily-Borel compactifications (available on Goresky's webpage).
In section 7 of this article, they work out the case of Siegel modular threefolds
in some detail. In particular, they show (formula 7.7.2)
that, if $x\in X_2$, then $IC_x$
has as a direct factor $H^*(X_\ell,\mathbb{C})$, where $X_\ell$ is a
locally symmetric space associated to the linear part of one of the maximal
Levi subgroups of $GSp(4)$; the linear part is in this case $GL(2)$. Morally, the cohomology of such "linear" locally symmetric spaces
is made up of Artin motives, so the factor $H^*(X_\ell,\mathbb{C})$ should be
of weight $0$ in every degree. But $H^1(X_\ell,\mathbb{C})$ contributes to $H^1(IC_x)$, so this vector space is
not pure of weight $1$.
Okay, now for the formal proof. Unfortunately, there is no result as precise
as the calculation of Goresky-Harder-MacPherson-Nair for the $\ell$-adic
intersection complex on the variety $X$ over $\mathbb{Q}$, but, if we reduce
modulo a prime of good reduction, then we can use the results of Morel's
article Complexes pondérés sur les compactifications de Baily-Borel :
le cas des variétés modulaires de Siegel. So now I will only consider the
reductions of the various varieties modulo $p$, where $p$ is a big enough
prime (here, it just means that $p$ does not divide the level).
I will need to introduce some notation. To make my life easier, I define
the symplectic group with a symplectic form such that the intersection of
$GSp(4)$ with the group of upper triangular matrices in $GL(4)$ is a Borel
subgroup of $GSp(4)$ (say, an antidiagonal form).
Let $N$ be the unipotent radical of this Borel
subgroup. Let $N_2$ be the center of $N$, so that $N_2$ is the intersection
of $GSp(4)$ and of the group
$\left(\begin{matrix}1 & 0 & * & * \\
0 & 1 & * & * \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1\end{matrix}\right)$
Let me also consider the following two cocharacters of $GSp(4)$ :
$w_1:\lambda\longmapsto \left(\begin{array}{cccc}\lambda^2 & 0 & 0 & 0 \\
0 & \lambda & 0 & 0 \\
0 & 0 & \lambda & 0 \\
0 & 0 & 0 & 1\end{array}\right)$
and
$w_2:\lambda\longmapsto \left(\begin{array}{cccc}\lambda & 0 & 0 & 0 \\
0 & \lambda & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1\end{array}\right)$
Note that the images of $w_1$ and $w_2$ normalize $N$ and $N_2$, so we
get two actions of $\mathbb{G}_m$ on $H^*(Lie(N),\mathbb{Q}_\ell)$ and
$H^*(Lie(N_2),\mathbb{Q}_\ell)$.
If $a$ and $b$ are integers, I will write
$H^*(Lie(N),\mathbb{Q}_\ell)_{\geq a,<b}$ for the part of $H^*(Lie(N),
\mathbb{Q}_\ell)$ where $w_2(\mathbb{G}_m)$ acts by characters $\lambda\longmapsto
\lambda^k$ with $k\geq a$ and $w_1(\mathbb{G}_m)$ acts by characters
$\lambda\longmapsto\lambda^k$ with $k<b$.
Likewise, I will write
$H^*(Lie(N_2),\mathbb{Q}_\ell)_{\geq a}$ for the part of $H^*(Lie(N_2),
\mathbb{Q}_\ell)$ where $w_2(\mathbb{G}_m)$ acts by characters $\lambda\longmapsto
\lambda^k$ with $k\geq a$. (I don't care about the other action.)
Let as before $x$ be a point of $X_2$. From theorem 4.2.1 of Morel's article,
I get a distinguished triangle :
$H^*(IC_x)\longrightarrow K_1\longrightarrow K_2,$
where $K_1$ is finite direct sum of complexes of the form
$H^*(\Gamma_\ell,H^*(Lie(N_2),\mathbb{Q}_\ell)_{\geq -3})$ and $K_2$ is a finite direct sum of complexes
$H^*(Lie(N),\mathbb{Q}_\ell)_{\geq -3,<-1}$. Here, $\Gamma_\ell$ is an
arithmetic subgroup of $GL(2,\mathbb{Q})$, where we see $GL(2)$ as a subgroup of
$GSp(4)$ via the map $g\longmapsto\left(\begin{array}{cc}g & 0 \\0 & {}^tg^{-1}\end{array}\right)$; then $GL(2)$ acts by conjugation on
$N_2$, hence it acts on $H^*(Lie(N_2),\mathbb{Q}_\ell)$, and it commutes
with $w_2(\mathbb{G}_m)$, so it stabilizes $H^*(Lie(N_2),\mathbb{Q}_\ell)_{\geq -3}$. Note that the arithmetic subgroup $\Gamma_\ell$ could vary in
the finite direct sum, but it won't matter.
(Note also that the theorem I cited only gives you an equality in a
Grothendieck group. However, if instead of Morel's 3.3.5 (that it relies
on), you instead invoke the stronger proposition 3.3.4 (ii), then you
can see that this equality comes from a distinguished triangle.)
Another thing we need to know is that the (Frobenius) weights on $K_1$ and
$K_2$ are given by minus the weights of $w_2(\mathbb{G}_m)$; that is,
if $w_2(\mathbb{G}_m)$ acts on some part of $K_1$ or $K_2$ by the character
$\lambda\longmapsto\lambda^k$, then the (Frobenius)
weight of this part is $-k$. (This is proved for example in an article
of Pink, and the reference is given in proposition 2.1.4 of the article
of Morel I'm citing.)
Now I need to look a little more closely at the action of my two tori on
the first cohomology groups of $Lie(N)$ and $Lie(N_2)$. They both act
trivially on the $H^0$ groups, so in particular $H^0(K_1)$
has (Frobenius) weight $0$ and $H^0(K_2)=0$ (because $H^0(Lie(N_2),\mathbb
{Q}_\ell)_{\geq -3}=H^0(Lie(N_2),\mathbb
{Q}_\ell)$ and $H^0(Lie(N),\mathbb
{Q}_\ell)_{\geq -3,<-1}=0$).
The torus $w_2(\mathbb{G}_m)$ acts by
$\lambda\longmapsto\lambda$ on $Lie(N_2)$, so it acts by $\lambda\longmapsto
\lambda^{-1}$ on $H^1(Lie(N_2),\mathbb{Q}_\ell)$, and $H^1(Lie(N_2),\mathbb{Q}_\ell)_{\geq -3}=H^1(Lie(N_2),\mathbb{Q}_\ell)$.
Let $V=N/N_2$; this is
a $1$-dimensional unipotent group. Using the spectral sequence
$E_2^{pq}=H^p(Lie(V),H^q(Lie(N_2),\mathbb{Q}_\ell))\Longrightarrow
H^{p+q}(Lie(N),\mathbb{Q}_\ell)$
and the fact that $H^p(Lie(V),\ )$ vanishes for $p>1$, we get an exact
sequence
$0\longrightarrow H^1(Lie(V),H^0(Lie(N_2),\mathbb{Q}_\ell))\longrightarrow
H^1(Lie(N),\mathbb{Q}_\ell)\longrightarrow H^0(Lie(V),H^1(Lie(N_2),\mathbb{Q}
_\ell))$.
On the third term, $w_1(\mathbb{G}_m)$ acts by $\lambda\longmapsto\lambda^k$
for $0\leq k\leq 2$ and $w_2(\mathbb{G}_m)$ acts by $\lambda\longmapsto
\lambda^{-1}$. On the first term, $w_1(\mathbb{G}_m)$ acts by $\lambda
\longmapsto\lambda^{-1}$ and $w_2(\mathbb{G}_m)$ acts trivially.
(This follows by looking at the weights of the tori in $Lie(N_2)$ and $Lie(V)$.)
In
particular, we find that $H^1(Lie(N),\mathbb{Q}_\ell)_{\geq -3,<-1}$
injects into $H^0(Lie(V),H^1(Lie(N_2),\mathbb{Q}_\ell))$, and this implies
that $H^1(K_2)$ has (Frobenius) weight $1$.
The distinguished triangle above gives us an exact sequence
$0=H^0(K_2)\longrightarrow H^1(IC_x)\longrightarrow H^1(K_1)\longrightarrow
H^1(K_2)$.
Remembering the formula for $K_1$ above, we see that $H^1(K_1)$ is a finite
direct sum of groups of the form $H^0(\Gamma_\ell,H^1(Lie(N_2),\mathbb{Q}_\ell))$
and $H^1(\Gamma_\ell,H^0(Lie(N_2),\mathbb{Q}_\ell))$ (with $\Gamma_\ell$
as above). The first summand has (Frobenius) weight $1$, and the second
summand has (Frobenius) weight $0$.
But we have seen that $H^1(K_2)$ has weight $1$. So the second summand is
sent to $0$ in $H^1(K_2)$, and $H^1(IC_x)$ contains a finite direct sum
of groups of the form $H^1(\Gamma_\ell,H^0(Lie(N_2),\mathbb{Q}_\ell))$,
in particular in contains a part of weight $0$, so it is not pure of weight $1$. (Note that all we did was
prove an algebraic version of a very small part of the result of
Goresky-Harder-MacPherson-Nair.)
This is really just an elaboration of Emerton's comment: You should read Mark Kisins' review of Faltings's paper "Almost etale extensions".
But I wanted to elaborate: Faltings regards the almost purity theorem as an analogue of Zariski-Nagata purity. In Faltings's original setup, it was formulated as follows. Consider the rings
$$ R_m = \mathbb{Z}_p[p^{1/p^m},T_1^{\pm 1/p^m},...,T_n^{\pm 1/p^m}]
$$
Each of them is smooth over $\mathbb{Z}_p[p^{1/p^m}]$ (in particular regular), and the transition maps are finite and etale after inverting $p$. Let $S_0$ be a finite normal $R_0$-algebra, which is etale after inverting $p$, and let $S_m$ be the normalization of $S_0\otimes_{R_0} R_m$. The idea is that there is ramification of $S_0$ in the special fibre, and you want to get rid of it, by adjoining the chosen tower of highly ramified rings $R_m/R_0$. It is not hard to see that this actually works almost at the generic point of the special fibre: At the generic point, the local ring is a discrete valuation ring, and the statement is that the discriminant of the extension of discrete valuation rings becomes arbitrarily small as $m\to\infty$ (this boils down to some more or less classical ramification theory).
Now, assume that you were lucky, and for some $m$, the ramification at the generic point of the special fibre is not just very close to zero, but actually zero on the nose. Zariski-Nagata purity tells you that the ramification locus of $S_m$ over $R_m$ has to be pure of codimension $1$, but you know that there is no ramification at the codimension $1$ points (which are either in characteristic $0$, or equal to the generic point of the special fibre) -- thus, there is no ramification at all, and $S_m$ over $R_m$ is etale.
The almost purity theorem says that this result extends to the almost world: In the limit as $m\to \infty$, $S_\infty$ over $R_\infty$ is almost etale (in the technical sense defined in almost ring theory).
Best Answer
You remember correctly, that the effect of Tate twisting the sheaf $\mathcal F$ is just to multiply the Frobenius eigenvalues by a factor of $q$. This result is not surprising because the Tate twist $\mathcal F(m)$ has no relation to the Serre twist on a projective variety, they are just denoted the same way.
The analogous operation to Tate twist in the classical cohomology of a complex variety is defined on the singular cohomology, not in coherent cohomology. There it has the effect of changing the natural $\mathbf Z$-structure on the complex cohomology by a factor $(2\pi i)$, shifting the weight filtration on the mixed Hodge structure by two steps and the Hodge filtration by one step.
Purity in $\ell$-adic cohomology has no direct relation to cohomological purity. But if you interpret a Frobenius eigenvalue of absolute value $q^{k/2}$ as something "k-dimensional", then both are statements that one thing or another is equidimensional, that it has pure dimension.