The answer to this depends on your comfort level with how color management works: How color spaces and color profiles affect the colors you see, and how comfortable you are resolving color mismatches related to color management. Yes, you might see unexpected color mismatches among different applications on a wide gamut (Adobe RGB or P3) display. It usually happens when some part of the workflow doesn’t fully support color management, like a free photo viewer program or some web browsers. If you have a good understanding of how color management works across various software and hardware, color mismatches on a wide gamut display are no mystery; you’ll know how to figure out where the problem is and how to address it.
If you aren’t yet comfortable with color spaces and color profiles, you’ll be more comfortable working on a display that reproduces the sRGB color gamut. Although sRGB is limited, and might not show you all of the colors in your images or your output device, color consistency among applications will generally be as reliable as you’re used to.
If you intend to improve your understanding of color management (for one thing, it’s a valuable professional skill to have), then I’d suggest getting an accurate pro display that can be switched between wide gamut and sRGB. (My NEC SpectraView supports multiple switchable color spaces.) You can start with the display set to sRGB for now, or when editing for media that are very sRGB-based such as web design. But you’ll be able to switch the display to wide gamut:
- When you want to see more colors than sRGB can reproduce
- As you learn more about using color management to handle different color spaces
- As you learn how to reconcile color-managed applications (like Lightroom) with non-color-managed applications
Keep in mind that while calibrating is essential, it doesn’t solve problems related to some applications being set differently (or incorrectly), or not supporting color management at all. That’s why people still see problems even though the calibrated their wide gamut display. They have to understand color management too.