The monitor will display perfectly at 1080p if that's the resolution you use for it, the internal screen resolution has absolutely no bearing on any external screen if used as a secondary/extended screen or if used as the only screen with the internal screen turned off. If you choose to do a clone/duplicate of the internal screen and keep it on then the internal screen will drop down resolution to 1920x1080 to match the external screen. Windows will not force a higher resolution onto a screen than it can support.
FYI, here's a 2K monitor for $129.99 after sales and rebates: http://www.newegg.com/Product/Product.aspx?Item=N82E16824025212
And a 4K for $289.99: http://www.newegg.com/Product/Product.aspx?Item=N82E16824025212
You should be able to find cheaper new or refurbished ones on sale today as well if you want a higher resolution screen.
FYI, here's a 2K monitor for $129.99 after sales and rebates: http://www.newegg.com/Product/Product.aspx?Item=N82E16824025212
And a 4K for $289.99: http://www.newegg.com/Product/Product.aspx?Item=N82E16824025212
You should be able to find cheaper new or refurbished ones on sale today as well if you want a higher resolution screen.
