This builds on the work by Alexander Mistakidis and Joyrex to support optional arrows and dropup menus. In my case I did not want to have an arrow on all of the dropdown menus, only some.
With this, you add the arrow class to the dropdown-menu element to get the arrow. If Bootstrap is positioning the dropdown/dropup to the left, also add arrow-right to shift the arrow to the other side.
// add an arrow to the dropdown menus
.dropdown-menu.arrow:before {
  position: absolute;
  left: 9px;
  display: inline-block;
  border-right: 7px solid transparent;
  border-left: 7px solid transparent;
  content: '';
}
.dropdown-menu.arrow:after {
  position: absolute;
  left: 10px;
  display: inline-block;
  border-right: 6px solid transparent;
  border-left: 6px solid transparent;
  content: '';
}
// postion at the top for a 'down' menu
.dropdown .dropdown-menu.arrow:before {
  top: -7px;
  border-bottom: 7px solid #ccc;
  border-bottom-color: rgba(0, 0, 0, 0.2);
}
.dropdown .dropdown-menu.arrow:after {
  top: -6px;
  border-bottom: 6px solid #ffffff;
}
// postion at the bottom for an 'up' menu
.dropup .dropdown-menu.arrow:before {
  bottom: -7px;
  border-top: 7px solid #ccc;
  border-top-color: rgba(0, 0, 0, 0.2);
}
.dropup .dropdown-menu.arrow:after {
  bottom: -6px;
  border-top: 6px solid #ffffff;
}
// support to move the arrow to the right-hand-side
.dropdown-menu.arrow.arrow-right:before,
.dropup .dropdown-menu.arrow.arrow-right:before {
  right: 15px;
  left: auto;
}
.dropdown-menu.arrow.arrow-right:after,
.dropup .dropdown-menu.arrow.arrow-right:after {
  right: 16px;
  left: auto;
}