05AB1E, 16 15 14 bytes
0ÐIv(2Fy+Š])Äà
Port of @tsh' top Python answer.
-1 byte thanks to @Neil, which also allowed a second byte to be saved
Try it online or verify all test cases.
Explanation:
0Ð # Push three 0s, let's call these a,b,c
I # Push the input-list
v # Pop and loop over each item `y`:
( # Negate the top item
2F # Loop 2 times:
y+ # Add the current integer `y` to the top
Š # Tripleswap x,y,z to z,x,y
# (after this inner loop, a,b,c is y+b,y-c,a)
] # Close both loops
) # Wrap all three values into a list
Ä # Get the absolute values of each
à # Pop and push the maximum