Depends what you mean by ‘support’. You can certainly put non-UCS-2 characters in a JS string using surrogates, and browsers will display them if they can.

But, each item in a JS string is a separate UTF-16 code unit. There is no language-level support for handling full characters: all the standard String members (length, split, slice etc) all deal with code units not characters, so will quite happily split surrogate pairs or hold invalid surrogate sequences.

If you want surrogate-aware methods, I'm afraid you're going to have to start writing them yourself! For example:

String.prototype.getCodePointLength= function() {
    return this.length-this.split(/[\uD800-\uDBFF][\uDC00-\uDFFF]/g).length+1;
};

String.fromCodePoint= function() {
    var chars= Array.prototype.slice.call(arguments);
    for (var i= chars.length; i-->0;) {
        var n = chars[i]-0x10000;
        if (n>=0)
            chars.splice(i, 1, 0xD800+(n>>10), 0xDC00+(n&0x3FF));
    }
    return String.fromCharCode.apply(null, chars);
};

I came to the same conclusion as bobince. If you want to work with strings containing unicode characters outside of the BMP, you have to reimplement javascript's String methods. This is because javascript counts characters as each 16-bit code value. Symbols outside of the BMP need two code values to be represented. You therefore run into a case where some symbols count as two characters and some count only as one.

I've reimplemented the following methods to treat each unicode code point as a single character: .length, .charCodeAt, .fromCharCode, .charAt, .indexOf, .lastIndexOf, .splice, and .split.

You can check it out on jsfiddle: http://jsfiddle.net/Y89Du/

Here's the code without comments. I tested it, but it may still have errors. Comments are welcome.

if (!String.prototype.ucLength) {
    String.prototype.ucLength = function() {
        // this solution was taken from 
        // http://stackoverflow.com/questions/3744721/javascript-strings-outside-of-the-bmp
        return this.length - this.split(/[\uD800-\uDBFF][\uDC00-\uDFFF]/g).length + 1;
    };
}

if (!String.prototype.codePointAt) {
    String.prototype.codePointAt = function (ucPos) {
        if (isNaN(ucPos)){
            ucPos = 0;
        }
        var str = String(this);
        var codePoint = null;
        var pairFound = false;
        var ucIndex = -1;
        var i = 0;  
        while (i < str.length){
            ucIndex += 1;
            var code = str.charCodeAt(i);
            var next = str.charCodeAt(i + 1);
            pairFound = (0xD800 <= code && code <= 0xDBFF && 0xDC00 <= next && next <= 0xDFFF);
            if (ucIndex == ucPos){
                codePoint = pairFound ? ((code - 0xD800) * 0x400) + (next - 0xDC00) + 0x10000 : code;
                break;
            } else{
                i += pairFound ? 2 : 1;
            }
        }
        return codePoint;
    };
}

if (!String.fromCodePoint) {
    String.fromCodePoint = function () {
        var strChars = [], codePoint, offset, codeValues, i;
        for (i = 0; i < arguments.length; ++i) {
            codePoint = arguments[i];
            offset = codePoint - 0x10000;
            if (codePoint > 0xFFFF){
                codeValues = [0xD800 + (offset >> 10), 0xDC00 + (offset & 0x3FF)];
            } else{
                codeValues = [codePoint];
            }
            strChars.push(String.fromCharCode.apply(null, codeValues));
        }
        return strChars.join("");
    };
}

if (!String.prototype.ucCharAt) {
    String.prototype.ucCharAt = function (ucIndex) {
        var str = String(this);
        var codePoint = str.codePointAt(ucIndex);
        var ucChar = String.fromCodePoint(codePoint);
        return ucChar;
    };
}

if (!String.prototype.ucIndexOf) {
    String.prototype.ucIndexOf = function (searchStr, ucStart) {
        if (isNaN(ucStart)){
            ucStart = 0;
        }
        if (ucStart < 0){
            ucStart = 0;
        }
        var str = String(this);
        var strUCLength = str.ucLength();
        searchStr = String(searchStr);
        var ucSearchLength = searchStr.ucLength();
        var i = ucStart;
        while (i < strUCLength){
            var ucSlice = str.ucSlice(i,i+ucSearchLength);
            if (ucSlice == searchStr){
                return i;
            }
            i++;
        }
        return -1;
    };
}

if (!String.prototype.ucLastIndexOf) {
    String.prototype.ucLastIndexOf = function (searchStr, ucStart) {
        var str = String(this);
        var strUCLength = str.ucLength();
        if (isNaN(ucStart)){
            ucStart = strUCLength - 1;
        }
        if (ucStart >= strUCLength){
            ucStart = strUCLength - 1;
        }
        searchStr = String(searchStr);
        var ucSearchLength = searchStr.ucLength();
        var i = ucStart;
        while (i >= 0){
            var ucSlice = str.ucSlice(i,i+ucSearchLength);
            if (ucSlice == searchStr){
                return i;
            }
            i--;
        }
        return -1;
    };
}

if (!String.prototype.ucSlice) {
    String.prototype.ucSlice = function (ucStart, ucStop) {
        var str = String(this);
        var strUCLength = str.ucLength();
        if (isNaN(ucStart)){
            ucStart = 0;
        }
        if (ucStart < 0){
            ucStart = strUCLength + ucStart;
            if (ucStart < 0){ ucStart = 0;}
        }
        if (typeof(ucStop) == 'undefined'){
            ucStop = strUCLength - 1;
        }
        if (ucStop < 0){
            ucStop = strUCLength + ucStop;
            if (ucStop < 0){ ucStop = 0;}
        }
        var ucChars = [];
        var i = ucStart;
        while (i < ucStop){
            ucChars.push(str.ucCharAt(i));
            i++;
        }
        return ucChars.join("");
    };
}

if (!String.prototype.ucSplit) {
    String.prototype.ucSplit = function (delimeter, limit) {
        var str = String(this);
        var strUCLength = str.ucLength();
        var ucChars = [];
        if (delimeter == ''){
            for (var i = 0; i < strUCLength; i++){
                ucChars.push(str.ucCharAt(i));
            }
            ucChars = ucChars.slice(0, 0 + limit);
        } else{
            ucChars = str.split(delimeter, limit);
        }
        return ucChars;
    };
}

More recent JavaScript engines have String.fromCodePoint.

const ideograph = String.fromCodePoint( 0x20001 ); // outside the BMP

Also a code-point iterator, which gets you the code-point length.

function countCodePoints( str )
{
    const i = str[Symbol.iterator]();
    let count = 0;
    while( !i.next().done ) ++count;
    return count;
}

console.log( ideograph.length ); // gives '2'
console.log( countCodePoints(ideograph) ); // '1'