Why does taking a Part fail in a Dataset query function when there is an operation in the index argument?
dataset = Dataset[Table[<|"a" -> n|>, {n, 3}]]; dataset[All, Range[4][[#a]]&] 
dataset[All, Range[4][[#a + 1]]&] 
To get it to work I need to use Extract instead:
dataset[All, Extract[Range[4], {#a + 1}]&] 
Are there other workarounds?
This behavior is concerned with Association's HoldAllComplete attribute.
Attributes[Association] {HoldAllComplete, Protected}
I am not completely certain of the internal workings but it appears that the a# + 1 expression is being held unevaluated when passed to Part. Wrapping the expression in Evaluate to force evaluation does not appear to help.
dataset[All, Range[4][[Evaluate[#a + 1]]] &] Message: Plus[1, Atom[Integer]] is not a valid part specification.
However, you there is a workaround by passing the expression as a parameter to a pure function.
dataset[All, Range[4][[#]] &@(#a + 1) &] Hope this helps.
How about
ds = Dataset[Table[<|"a" -> n|>, {n, 3}]]; ds[All, "a", # + 1 &] 
BTW, you 1st query can be simplified to
ds[All, "a"]