Add
html,body{min-height:100%}
to your CSS.
As for the doctype, use this
<!DOCTYPE HTML>
so your document will look somewhat like this working example. Try it and you'll see the image is displayed just the way you want it to be displayed.
<!DOCTYPE HTML>
<html>
<head>
<title>Title Here</title>
<meta charset="utf-8">
<style>
html{
min-height:100%;
}
body{
min-height:100%;
background-color: #77d5fb;
background-image: url('bottom_bg.jpg');
background-position: center bottom;
background-repeat: no-repeat;
font-family: Comfortaa;
}
#searchbar{
width:700px;
height:200px;
position:absolute;
left:50%;
top:50%;
margin:-100px 0 0 -350px;
}
</style>
</head>
<body>
<p id="searchbar">Just testing!</p>
</body>
</html>
Enjoy!