It's not intrinsic to height maps that they stair step. It's because you use 8-bits which only offers 256 levels. People use color maps, but put the same gray level in all three channels, so its no better than an 8-bit gray map.

Use a 16-bit map gray map instead. Or a 32-bit-EXR or HDR. Then you will see no stair steps.

Normals have 24 bits per pixel. If you gave 24-bits to height maps they'd be just as good.

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)