Longest path in grid like graph
If all numbers are distinct then you can find the longest path using straightforward dynamic programming. The general problem is probably NP-complete, since deciding whether a grid graph has a Hamilton path is NP-complete, as shown by Itai, Papadimitriou and Szwarcfiter. Grid graphs are finite induced subgraphs of the infinite rectangular grid, in which non-diagonal cells are not adjacent; however it seems reasonable to conjecture that Hamilton path is NP-complete also for finite subgraphs of the infinite rectangular grid with diagonals. Given that, it is likely that your problem is also NP-complete.