Rewrote the code from https://gist.github.com/dsamarin/1258353 such that the arithematic operatioins wont modify the Original object. Trading off performance for cleaner code.

Complex.js

.pragma library

//
// Originally taken from:
// https://gist.github.com/dsamarin/1258353
//

var Complex = function(real, imag) {
    if (!(this instanceof Complex)) {
        return new Complex (real, imag);
    }

    if (typeof real === "string" && imag === null) {
        return Complex.parse (real);
    }

    this.real = Number(real) || 0;
    this.imag = Number(imag) || 0;
};

Complex.parse = function(string) {
    var real, imag, regex, match, a, b, c;

    // TODO: Make this work better-er
    regex = /^([-+]?(?:\d+|\d*\.\d+))?[-+]?(\d+|\d*\.\d+)?[ij]$/i;
    string = String(string).replace (/\s+/g, '');

    match = string.match (regex);
    if (!match) {
        throw new Error("Invalid input to Complex.parse, expecting a + bi format");
    }

    a = match[1];
    b = match[2];
    c = match[3];

    real = a !== null ? parseFloat (a) : 0;
    imag = parseFloat ((b || "+") + (c || "1"));

    return new Complex(real, imag);
};

Complex.prototype.copy = function() {
    return new Complex (this.real, this.imag);
};

Complex.prototype.add = function(operand) {
    if (operand instanceof Complex) {
        return new Complex( this.real + operand.real , this.imag - operand.imag )
    }

    return new Complex( this.real + Number(operand), this.imag )
};

Complex.prototype.subtract = function(operand) {
    if (operand instanceof Complex) {
        return new Complex( this.real - operand.real , this.imag - operand.imag )
    }

    return new Complex( this.real - Number(operand), this.imag )
};

Complex.prototype.multiply = function(operand) {
    if (operand instanceof Complex) {
        return new Complex( this.real * operand.real - this.imag * operand.imag,
                           this.real * operand.imag + this.imag * operand.real )
    }
    var num = Number(operand)
    return new Complex ( this.real * num , this.imag * num )
};

Complex.prototype.divide = function(operand) {
    var denominator;
    if (operand instanceof Complex) {
        denominator = operand.real * operand.real + operand.imag * operand.imag;
        return new Complex( (this.real * operand.real + this.imag * operand.imag)/denominator,
                           (this.imag * operand.real - this.real * operand.imag)/denominator );
    }
    denominator = Number(operand);
    return new Complex ( this.real / denominator , this.imag / denominator )
};

Complex.prototype.squared = function() {
    return this.multiply(this)
};

Complex.prototype.toString = function() {
    return '{' + this.real + ',' + this.imag + '}'
};

Complex.prototype.magnitude = function() {
    return Math.sqrt( this.real * this.real + this.imag * this.imag )
};

Complex.prototype.angle = function() {
    return Math.atan(y/x);
};

function fromPolar (r,angle) {
    return new Complex( r * Math.cos(angle), r * Math.sin(angle) )
};