Creating custom JSON object

• You had some Property-naming issues with camelCased "depType". Fix that.
• Create a new copy of the desired object to manipulate using JSON.parse(JSON.stringify(orgObj))
• Loop that object to find the desired Property "Department"
• Since Department is an Array of Objects you need to loop that Array for(var i=0; i<dep.length; i++).
• Than you'll need to match if that Array contains arrObj.hasOwnProperty( "depType" )
• if successful you can than fill your new object with all the info newObj[arrObj.depType] = arrObj;
• Since now, inside your new object there's also the good old "depType" property you can get rid of it using delete.

jsBin demo

var myjson = {
  "Department": [
      {
        "depType": "Testing",   // NOTE: "camelCase"
        "name": "xyz",
        "address":""
      },
      {
        "depType": "Developer", // FIX: "camelCase" !!
        "name": "abc"
      }
    ]
};

function depTypify( orgObj ) {
  var objCopy = JSON.parse(JSON.stringify(orgObj)); // Fresh copy
  var newObj = {};
  for(var prop in objCopy){
    if(prop === "Department") {
      var dep = objCopy[prop];          // get Department Array
      for(var i=0; i<dep.length; i++) { // Loop array
        var arrObj = dep[i];            // Explore Each Array Object
        if(arrObj.hasOwnProperty( "depType" )) {
          newObj[arrObj.depType] = arrObj;
          delete arrObj.depType;        // We don't need it any more
        }
      }
    }
  }
  return newObj;
}

var myNewJson = depTypify( myjson );

if you do than console.log( myNewJson ) this is what you'll get:

[object Object] {
  Developer: [object Object] {
    name: "abc"
  },
  Testing: [object Object] {
    address: "",
    name: "xyz"
  }
}

The nice thing is that your old json is still intact.