There is the tool "Vertical Distance to Channel Network" in SAGA GIS (open source). SAGA is also connected to QGIS allowing you to execute this tool directly from QGIS. Look for the tool in the terrain analysis/ channels toolbox.

You could also do it in two steps (in ArcGIS or QGIS): first create a map in which every grid cell has te elevation of the nearest channel (nearest neighbor), and next subtract this map from you elevation map.

@Florian Betz, thank you for your answer, it works now with SAGA GIS.

@Judith I will try

For GRASS GIS, there is the addon "r.stream.distance" to calculate distance to streams and outlets and elevation above streams and outlets. In outlets mode it can also calculate parameters for subbasins. See

https://grasswiki.osgeo.org/wiki/R.stream.*_modules#r.stream.distance

and

https://grass.osgeo.org/grass70/manuals/addons/r.stream.distance.html

https://grasswiki.osgeo.org/wiki/R.stream.*_modules#r.stream.distance

Thank you for your answer