While Apple (AAPL -0.81%) is notoriously secretive, its management is still willing to drops hints about forthcoming products from time to time. Apple's CEO Tim Cook has, for example, never said no to a bigger iPhone and often speaks positively about the opportunities in TV, wearable devices, and mobile payments.

But one thing Cook and other Apple executives have consistently ruled out is a hybrid operating system -- a convergence of OS X and iOS that would unite Apple's product lines in a similar way to Microsoft's (MSFT -0.66%) Windows 8.

But analysts won't stop talking about it. Last week, JPMorgan Chase revived the rumor in a note predicting a forthcoming "iAnywhere" -- an operating system that would merge Apple's iPad line with its Mac PCs. Based on comments from Apple's leaders, I strongly doubt such a product will ever appear. But if it does, it should have an immense on the entire industry, and accessory suppliers such as Logitech (LOGI -0.87%).

JPMorgan calls for "iAnywhere"
JPMorgan analysts believe Apple is working on a converged platform, a new operating system that would combine iOS's mobile, touch-based capability with OS X's desktop, productivity capability.

In a hypothetical example, you take your iPhone and plug it into a special dock that connects to an Apple-made display, keyboard, and mouse. The iPhone then switches to OS X mode, allowing you to do the sort of work you would've done on an iMac.

Apple wouldn't be the first company to experiment with such an idea. Motorola tried something similar with the Atrix phone years ago, and the failed Kickstarter Ubuntu Phone project promised a Linux-based phone that could transform into a desktop computer. Asus' Padfone is in a similar vein.

Apple's management has consistently downplayed hybrids
The obvious problem with such a notion is that Apple, as a company, is based entirely around selling physical stuff. Yes, iTunes has grown significantly in recent years, but it remains a tiny fraction of Apple's overall revenue. JPMorgan claims there would be an opportunity in selling docks and accessories, but ultimately, Apple would probably prefer its customers bought an iPhone, iPad, and iMac, rather than just an iPhone and a couple of docks.

More concretely, Apple's management has explicitly ruled out a hybrid operating system on numerous occasions. As far back as 2012, Cook openly mocked Microsoft's Windows 8 as an attempt to converge a toaster with a refrigerator, and then late last year, characterized Apple's competitors (read: Microsoft) as "confused."

Craig Federighi, Apple's SVP of software rngineering, told Macworld last month that operating-system convergence was a "nongoal," and that Apple had no intention of turning iOS into OS X, or OS X into iOS.

I suppose that Apple could get around this by running two completely separate operating systems on the same device, shifting back and forth between them based on how the device is being used, similar to the growing number of tablet hybrids that run both Microsoft's Windows and Google's Android. Still, it just doesn't seem likely, given how much Apple's executives have explicitly criticized the very notion of a hybrid product.

Microsoft would seem like the more obvious choice
If the concept does have any merit, then it's more likely to benefit Microsoft -- a company that's explicitly moving in that direction.

More than just a hybrid operating system, the overarching philosophy behind Windows 8, repeated in every Microsoft commercial, is "one experience for everything in your life." Microsoft already sells a docking station for its Surface Pro, allowing the laptop/tablet hybrid to easily transform into a desktop PC.

And it will likely push this philosophy further with the next version of Windows, codenamed Threshold. Microsoft blogger Paul Thurrott has said that the next version of Windows will put Microsoft on "the threshold ... of a truly consolidated OS."

Logitech will either be a big winner or a total loser
From an investing perspective, the stock that could be most affected by Apple's push into a converged operating system is Logitech. The PC component supplier has been a great small cap to own, rallying more than 120% in the last six months.

Much of that gain has come from a string of solid earnings reports fueled by Logitech's success in tablet accessories. While sales of Logitech's PC keyboards have been stagnant, it has seen tremendous growth in tablet keyboards.

Logitech sells keyboards for several different tablet vendors, but most of its products are aimed at Apple's iPads. In the event that Apple were to commit to making its own accessories, it could eliminate much of  the demand for Logitech's products. On the flipside, however, there's clearly a market for these products, and if Apple doesn't take advantage of it, Logitech will.

iAnywhere the next Apple catalyst?
Based on the philosophy Apple's management has consistently articulated, I strongly doubt that the company would move to offer a converged platform. It certainly could, but it seems highly unlikely. However, Microsoft is very explicitly moving in that direction, and if that's the way the computing market is headed, Microsoft is the company that would benefit the most.