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DendogramPlot accepts Axes as an option. Despite syntax highlighting in red of Axes and AxesOrigin, GridLines etc. these options seem to work with DendogramPlot.

DendrogramPlot accepts Axes as an option. Despite syntax highlighting in red of Axes and AxesOrigin, GridLines etc. these options seem to work with DendrogramPlot.

In the following, these distances are highlighted in red:

Needs["HierarchicalClustering`"]
Grid[{{Agglomerate[{1, 2, 10, 4, 8},
 DistanceFunction -> Automatic, 
 Linkage -> "Single"]}, 
 {DendrogramPlot[{1, 2, 10, 4, 8},
 DistanceFunction -> Automatic, Linkage -> "Single", 
 LeafLabels -> (# &), ImageSize -> 300, Axes -> {False, True}, 
 AxesOrigin -> {0, Automatic}]}}]

Few more combinations where inter-cluster distances are highlighted in red and shown as green gridlines in the dendogram plot:

 Grid[{{ToString@#[[1]] <> "--" <> #[[2]]}, {Replace[
   Agglomerate[{1, 2, 10, 4, 8},
    DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
   Cluster[a_, b_, c_, d__] -> 
    Cluster[a, b, Style[c, 18, Red, Bold], d], {0, 
    Infinity}]}, {DendrogramPlot[{1, 2, 10, 4, 8},
   DistanceFunction -> #[[1]], Linkage -> #[[2]], 
   LeafLabels -> (# &), 
   GridLines -> {None, Cases[Agglomerate[{1, 2, 10, 4, 8},
       DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
      Cluster[a_, b_, c_, d__] :> c, {0, Infinity}]}, 
   GridLinesStyle -> Green, ImageSize -> 500, 
   Axes -> {False, True}, AxesOrigin -> {.75, Automatic}]}}] & /@ 
 Tuples[{{Automatic, ManhattanDistance}, {"Complete",  "Centroid"}}] // Column

EDIT: Note: Despite syntax highlighting in red of Axes and AxesOrigin, the options seem to work:

Several combinations of DistanceFunction and Linkage where inter-cluster distances are highlighted in red and shown as green gridlines in the dendogram plot:

Needs["HierarchicalClustering`"]

Grid[{{ToString@#[[1]] <> "--" <> #[[2]]}, 
  {Replace[ Agglomerate[{1, 2, 10, 4, 8},
    DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
    Cluster[a_, b_, c_, d__] -> 
    Cluster[a, b, Style[c, 18, Red, Bold], d], {0, 
    Infinity}]}, {DendrogramPlot[{1, 2, 10, 4, 8},
   DistanceFunction -> #[[1]], Linkage -> #[[2]], 
   LeafLabels -> (# &), 
   GridLines -> {None, Cases[Agglomerate[{1, 2, 10, 4, 8},
       DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
      Cluster[a_, b_, c_, d__] :> c, {0, Infinity}]}, 
   GridLinesStyle -> Green, ImageSize -> 500, 
   Axes -> {False, True}, AxesOrigin -> {.75, Automatic}]}}] & /@ 
 Tuples[{{Automatic, ManhattanDistance}, {"Complete",  "Centroid"}}] // Column

EDIT: What I get for Frederik's example in the comments:

DendogramPlot accepts Axes as an option.

Inter-cluster distances in a Cluster object are given as the third from last element.

So ... this verifies that vertical axis does indeed measure the inter-cluster distances for a given DistanceFunction and Linkage.

DendogramPlot accepts Axes as an option. Despite syntax highlighting in red of Axes and AxesOrigin, GridLines etc. these options seem to work with DendogramPlot.

Inter-cluster distance in a Cluster object is given as the third element.

Few more combinations where inter-cluster distances are highlighted in red and shown as green gridlines in the dendogram plot:

 Grid[{{ToString@#[[1]] <> "--" <> #[[2]]}, {Replace[
   Agglomerate[{1, 2, 10, 4, 8},
    DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
   Cluster[a_, b_, c_, d__] -> 
    Cluster[a, b, Style[c, 18, Red, Bold], d], {0, 
    Infinity}]}, {DendrogramPlot[{1, 2, 10, 4, 8},
   DistanceFunction -> #[[1]], Linkage -> #[[2]], 
   LeafLabels -> (# &), 
   GridLines -> {None, Cases[Agglomerate[{1, 2, 10, 4, 8},
       DistanceFunction -> #[[1]], Linkage -> #[[2]]], 
      Cluster[a_, b_, c_, d__] :> c, {0, Infinity}]}, 
   GridLinesStyle -> Green, ImageSize -> 500, 
   Axes -> {False, True}, AxesOrigin -> {.75, Automatic}]}}] & /@ 
 Tuples[{{Automatic, ManhattanDistance}, {"Complete",  "Centroid"}}] // Column

So ... vertical axis does indeed measure the inter-cluster distances for a given DistanceFunction and Linkage.