Just steer clear of nested loops (O(n^2)), and .indexOf() (+O(n)).

function merge(a, b) {
    var hash = {}, i;
    for (i=0; i<a.length; i++) {
        hash[a[i]]=true;
    } 
    for (i=0; i<b.length; i++) {
        hash[b[i]]=true;
    } 
    return Object.keys(hash);
}

Just steer clear of nested loops (O(n^2)), and .indexOf() (+O(n)).

function merge(a, b) {
  var hash = {};
  var i;
  
  for (i = 0; i < a.length; i++) {
    hash[a[i]] = true;
  }
  for (i = 0; i < b.length; i++) {
    hash[b[i]] = true;
  }
  return Object.keys(hash);
}

var array1 = ["Vijendra", "Singh"];
var array2 = ["Singh", "Shakya"];

var array3 = merge(array1, array2);

console.log(array3);