I want to change color of caret and make it not relative from input text, because it's hard to make it look good when it is.
There are two ways I have done this. If you want this to affect all carets on your site (probably not preferred) then you could override the caret css class in you:
.caret{border-top:4px solid red;}
Another option is to create a custom class and add it to the caret in your markup
.red{border-top:4px solid red;}
<div class="btn-group">
<button type="button" class="btn btn-default dropdown-toggle" data-toggle="dropdown">Action<span class="caret red"></span></button>
<ul class="dropdown-menu" role="menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li><a href="#">Separated link</a></li>
</ul>
</div>
The original JSFiddle has been updated http://jsfiddle.net/whoiskb/pE5mQ/