Is it possible to create custom operators in JavaScript?

The short answer is no. ECMAScript (the standard JS is based on) does not support operator overloading.

As an aside, in ECMAScript 7, you'll be able to overload a subset of the standard operators when designing custom value types. Here is a slide deck by language creator and Mozilla CTO Brendan Eich about the subject. This won't allow arbitary operators, however, and the overloaded meaning will only be applied to value types. <- haha that ended up not happening.

It is possible to use third party tools like sweet.js to add custom operators though that'd require an extra compilation step.

I've answered with a solution from outside JavaScript using esprima - this is changing JavaScript and extending it, it's not native.

Given the somewhat new tagged template literals feature that was added in ES6 one can create custom DSLs to handle embedded expressions such as these including different algebraic symbols.

ex. (run in stackblitz)

function math(strings, x, y) {
  // NOTE: Naive approach as demonstration

  const operator = strings[1].replace(/\s/gi, "");

  if (operator == "∘") {
    return x + 2 * y;
  }
  else if (operator == "^") {
    return Math.pow(x, y);
  }
  else {
    return `Unknown operator '${operator}'`;
  }
}

console.log(math`${2} ∘ ${2}`)

Note that since tagged template literals don't necessarily return strings as results they can return more complex intermediate AST like structures to build up an expression that can then be further refined and then interpreted while keeping close to the domain specific language at hand. I haven't found any existing library that does this yet for Javascript but it should be an interesting and quite approachable endeavor from what it appears from what I know of tagged template literals and usage in such places as lit-html.