I think Cortana should be a platform (and operating system) for emerging smart devices, i.e. wearables, appliances, smart TVs, cars and other 'internet of things' devices.
Making Cortana ubiquitous across these devices could allow for (1) de-facto support for Windows Phone, Windows and Xbox and (2) an actual use-case for a Cortana app on iOS and Android. For example, in order to make full use of a smartwatch I need to tie it the Cortana app on my iPhone/Galaxy.
Of course, these are short-term achievements, making Cortana the springboard for development on these new systems can cement for Microsoft a dominant position in the 'internet of things' world. In other words, more developers working with Microsoft tools and languages.
Why should this be the route? Imagine buying smart furniture, e.g. desk for your office and mirror for your bathroom, and having the option of having it loaded with Cortana instead of Chrome OS.