WebExtensions are likely to be The One True API. Chrome, Opera, Edge, and Firefox all support them, and we're all working on standardizing them at the W3C: https://www.w3.org/community/browserext/
> If I wanted handicapped extension APIs, I'd just use Chrome.
Though we're trying to avoid reinventing wheels, we're not limiting ourselves to Chrome's APIs. For example, https://bugzil.la/1242871 extends Chrome's webRequest API to provide additional metadata needed by the NoScript and RequestPolicy add-ons. The initial development is focused on Chrome parity, since that covers most add-ons, but once we get there you'll see more of an emphasis on landing APIs that are needed by existing, more powerful Firefox add-ons.
If you're an add-on developer yourself and are concerned about APIs you need, please fill out this survey so we have a record of it: https://docs.google.com/a/mozilla.com/forms/d/1PtRXcs9UHVMjg...