Is it possible to use multiple compute buffers in compute shader in HLSL in Unity?

You can define a RWStructuredBuffer - RW means Read+Write A normal StructuredBuffer is write-protected: you can't write TO it from the shader, but read it after passing it from C#.

You can define multiple in shader:

RWStructuredBuffer<float4> myFirstBuffer : register(u1); // index1 !
RWStructuredBuffer<float4> mySecondBuffer : register(u2); // index2 !

And access them like this in C#:

Graphics.ClearRandomWriteTargets();
material.SetPass(0);
material.SetBuffer("myFirstBuffer", myFirstBuffer);
Graphics.SetRandomWriteTarget(1, myFirstBuffer, false); // index 1!
compute_buffer.GetData(myFirstBuffer);

material.SetBuffer("mySecondBuffer", mySecondBuffer);
Graphics.SetRandomWriteTarget(2, mySecondBuffer, false); // index 2!
compute_buffer.GetData(mySecondBuffer);