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?