Extract all keys from an object (json)

JavaScript (V8), 72 bytes

An edited version to support literal false, true and null values.

f=(o,s)=>!o|[o]==o||Object.keys(o).map(k=>f(o[k],k=s?s+[,k]:k,print(k)))

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JavaScript (V8), 69 bytes

Takes a native JSON object as input. Prints the results, using a comma as the delimiter.

f=(o,s)=>[o]==o||Object.keys(o).map(k=>f(o[k],k=s?s+[,k]:k,print(k)))

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How?

This is a recursive function walking through the keys at the root level and then in each sub-tree of the structure.

We need to process recursive calls on objects and arrays and to stop on strings and numbers. This is achieved with [o]==o||Object.keys(o):

 type of o | [o]==o    | Object.keys(o)  | string coercion example
-----------+-----------+-----------------+-------------------------------------
 array     | false     | 0-based indices | ['foo', 'bar'] -> 'foo,bar'
 object    | false     | native keys     | {abc: 'xyz'}   -> '[object Object]'
 string    | true      | n/a             | 'hello'        -> 'hello'
 number    | true      | n/a             | 123            -> '123'

Ruby, 108 146 115 92 bytes

+38 bytes to fix test cases for objects inside arrays.....

-31 bytes because we can take a parsed JSON object as input now.

-17 bytes by removing the duplicated flat_map usage.

f=->j,x=''{z=j==[*j]?[*0...j.size]:j.keys rescue[];z.flat_map{|k|[r=x+k.to_s]+f[j[k],r+?.]}}

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JavaScript (Node.js), 75 bytes

f=o=>Object.keys(o+''===o||o||0).flatMap(k=>[k,...f(o[k]).map(i=>k+'.'+i)])

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