<div class="row">
    <div class="col-xs-12 col-md-6">
      <input type="text" placeholder="Base input">
    </div>
    <div class="col-xs-12 col-md-6">
      <div>
        <input type="text" placeholder="Dropdown-like input">
        <label class="dropdown-arrow-box"><i class="fa fa-chevron-down dropdown-arrow"></i></label>
      </div>
    </div>
  </div>

input[type=text] {
  box-sizing: border-box;
  display: inline-block;
  padding: 12px 20px;
  margin: 8px 0;
  border: 2px solid white;
  color: white;
  float: left;
  font-size: 1.5rem;
  font-weight: 400;
  height: 60px;
  vertical-align: middle;
}

.dropdown-arrow-box {
  background-color: white;
  margin: 8px 0;
  border: 2px solid white;
  box-sizing: border-box;
  display: inline-block;
  font-size: 2.5rem;
  line-height: 60px;
  height: 60px;
  text-align: center;
  vertical-align: middle;
  width: 60px;
}

.dropdown-arrow {
  color: #0db0a7;
}