Image: Microsoft.

Even before Cortana was available on Windows Phone, there were discussions about whether Microsoft's digital assistant would come to other devices.

The latest rumour that she will be available on iOS and Android might restart the debate about whether Microsoft should save key services for its own platforms. But actually, bringing Cortana to other smartphones makes it an even better feature on Windows 10 - and not just because more users issuing more queries means more raw data for Bing to mine and improve Cortana with.

Microsoft's cross-platform strategy isn't just that if you have Office or Skype or any other Microsoft service everywhere, then you're less likely to switch to another service. It's that if you don't have the service you use on every device you pick up, then you're not truly mobile, even on the devices that do have it.

The Microsoft slogan of the moment is "mobility of experience". Microsoft CEO Satya Nadella used the term at the Windows 10 event in January.

"When we talk about mobility, it is not about the mobility of any single device, but it is the mobility of the experiences across devices," he said. "Windows 10 is built for a world where there are going to be more devices on the planet than people. That means the mobility of the experience is what matters, not the mobility of the device."

Stephen Elop echoed him almost word for word at Mobile World Congress: "We should not obsess on the mobility of any single device, but we should focus on the mobility of the experiences a person has across devices."

But it was Terry Myersom, Microsoft's executive vice president of operating systems, who actually explained what "mobility of experience" means. "The number of smart devices that [are owned by] families, ourselves, or in the workplace is just exploding all around us. It should be easy to put one device down and pick another up, pick another and continue where you left off. The technology needs to get out of the way and provide mobility of the experience."

He was talking about Windows 10, but it's how Office already works across devices. Start a document on one device, save it in OneDrive, and it's on the recent files list when you open Office on your iPad or your Mac or your Android phone.

In the same way, having Cortana on your iPhone would make her more useful in Windows 10 - and might even make you upgrade or buy a Windows 10 PC - not just to get the same experience but to get an experience that's better because you have it on both devices. Reminders are the obvious example. Set a reminder on your PC to pick up eggs next time you pass the convenience store, and you really want Cortana on the phone in your pocket to set off that reminder as you're passing the shop.

Given how much searching we do on phones, having Cortana (with your permission) sync the searches you do in the Spartan browser means you can find not just the page you were looking at on your other device - which IE can already do - but how you found it.

It could even help Cortana get access to your information without needing the same level of integration as on Windows 10 and Windows Phone.

For Cortana to tell you when to leave for a meeting to beat traffic, the service needs to see your calendar. To spot which flights to track for you, it needs to check your email. Getting locations so Cortana can work out what's home and what's the office and where your favourite coffee shop is shouldn't be a problem, but getting other information on an iPhone or Android device might be harder. If Cortana can get that information from your Windows PC though, she can know what you need and tell you about it on any device.

Cortana currently stores information about your interests in a notebook on the device. Move that into the cloud and Cortana can spot emails telling you the tracing number of your package on your PC and alert you on your phone that the courier should be showing up in the next hour. That has to be carefully done to protect your privacy - something the Cortana team is very keen on. But if Microsoft gets Cortana on iOS and Android right, that could turn into a way of getting those users on a Windows 10 PC as well.

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