Does this do what you have in mind?

    pts = {{1, 1}, {1, 2}, {1234512345, 1234512345}, {1234512345, 1234512346}};
    edges = Join @@ MapIndexed[
        Thread[{#1, #2[[1]]}] &,
        Nearest[
          pts -> "Index", 
          pts, 
          {\[Infinity], 1}, 
          DistanceFunction -> ManhattanDistance
        ][[All, 2 ;;]]
      ];
    SparseArray`StronglyConnectedComponents[SparseArray[edges ->_]]

For diagonal connectivity the same code with `ManhattanDistance` replaced by `ChessboardDistance` should work.