There’s a Moment.js plugin on NPM called frozen-moment – You could use moment().freeze() in place of Object.freeze(moment()).
Otherwise, vanilla Moment.js has a clone method that should help you avoid mutability problems, so you could do something like this:
var a = moment(),
b = a.clone(); // or moment(a)
UPDATE:
It has been two years since I wrote this answer. In this time, another library for working with dates has surfaced and gained a lot of traction: https://date-fns.org/
This library is immutable by default and follows a modular, functional architecture, which means it is better suited to tree shaking and client-side bundling. If you are working on a project that makes extensive use of Webpack on the client side, and find that Moment.js is giving you trouble with your build, or even if Moment.js’ mutability is causing you a lot of headache, then you should give date-fns a try.