The lines that use the (?) operator are compiler errors. I suspect this fails for the same reason that you cannot add operators in type extensions. I just don't know what that reason is...

getOperator.fs

[<AutoOpen>]
module Auto =
  type Result<'s, 'f> =
  | Success of 's
  | Failure of 'f
  open System
  open System.Runtime.CompilerServices
  open System.Collections.Generic
  open Newtonsoft.Json.Linq

  type JObject with
    member inline this.Get (n: string) =
      let l = n.ToLowerInvariant()
      let matchKey (p: JProperty) = p.Name.ToLowerInvariant() = l
      match this.Properties() |> Seq.tryFind matchKey with
      | Some o -> Some <| o.Value.Value<'t>()
      | None -> None

  [<Extension>]
  type Extensions () =
    [<Extension>]
    static member inline Get (i: IDictionary<string,Object>, n: string) =
      let l = n.ToLowerInvariant()
      let matchKey (KeyValue(n': string, o)) = n'.ToLowerInvariant() = n
      match i |> Seq.tryFind matchKey with
      | Some (KeyValue(k,v)) -> Some (v :?> 't)
      | None -> None
    

  let inline (?) (x: ^a) (n: string) : Result<'t,string> =
    try
    match (^a : (member Get : string -> 't option) (x, n)) with
    | Some v -> Success v
    | None -> Failure n
    with _ -> Failure n

  let foo () =
    let d = dict [||]
    d?foo |> ignore
    let j = JObject()
    j.Get("foo") |> ignore
    j?foo |> ignore