Its very easy to remove next element by checking class is matches or not. You can also remove previous element as well as next element.
$(".base").prev(".search").remove();
<script src="3.2.1/jquery.min.js"></script>
<div class="base">ELEMENT 1</div>
<div class="will_not_remove">ELEMENT WILL NOT REMOVE</div>
<div class="remove">HAS CLASS "REMOVE" BUT WILL NOT REMOVE</div>
<div class="base">ELEMENT 2</div>
<div class="remove">ELEMENT WILL REMOVE</div>
<script type="text/javascript">
$(".base").next(".remove").remove();
</script>
$(".base").prev(".search").remove();
<script src="3.2.1/jquery.min.js"></script>
<div class="base">ELEMENT 1</div>
<div class="will_not_remove">ELEMENT WILL NOT REMOVE</div>
<div class="remove">HAS CLASS "REMOVE" BUT WILL NOT REMOVE</div>
<div class="base">ELEMENT 2</div>
<div class="remove">ELEMENT WILL REMOVE</div>
<script type="text/javascript">
$(".base").next(".remove").remove();
</script>