square squares with diagonals also squares The numbers reading across and down in these squares are square:
$\begin{array}{ccc}
1 & 4 & 4\\
4 &8&4\\
4&4&1
\end{array}$
$\begin{array}{ccc}
5&2&9\\
2&5&6\\
9&6&1
\end{array}$
$\begin{array}{cccc}
2&1&1&6\\
1&2&2&5\\
1&2&9&6\\
6&5&6&1
\end{array}$
Are there any such square squares where the diagonals are also squares? If not in base 10, is it possible in other bases?

Those matrices are certainly hard to find, I tried it a lot and found that there are no $4\times4$ or $3\times3$ matrices that do what you want in base $10$ or below. However watch this one in base $11$:
$$\begin{array}{ccc}
1 & 9 & 5\\
9 & 6 & 1\\
5 & 1 & 9
\end{array}$$
You have
$169_{11}=196=14^2$
$195_{11}=225=15^2$
$519_{11}=625=25^2$
$565_{11}=676=26^2$
$961_{11}=1156=34^2$
All colums, rows, diagonals when read in any way (colums and rows only down and right) are squares, woha ;-)
