The latest HDMI 2.1 adds native support for 120Hz at 4K UHD and 60Hz at 8K. HDMI 2.0 is also fairly standard and can be used for 240Hz at 1080p, 144Hz at 1440p and 60Hz at 4K. So, make sure the exact monitor you’re looking at supports 144Hz or at least 120Hz over its HDMI 1.4 port - or just look for a 144Hz display with HDMI 2.0.
In fact, most 144Hz monitors are limited to 120Hz over HDMI 1.4, but some monitor manufacturers will utilize custom timing parameters, pixel rates, etc to use the full potential of the HDMI 1.4 bandwidth, thus getting 144Hz at 1080p. However, keep in mind that some monitors have limited bandwidth over HDMI 1.4 making it max out at 120Hz or even at 60Hz in some cases (mostly G-SYNC and older monitors). Nowadays, most monitors, TVs and graphics cards feature at least an HDMI 1.4 port which is enough for 144Hz at 1080p, 75Hz at 1440p and 30Hz at 4K. Only the Dual-Link DVI-D port can do 1920×1080 at 144Hz or 2560×1440 at 75Hz, while other DVI ports max out at the same bandwidth as VGA. DVI (Digital Visual Interface) is a more common and widespread connector, but bear in mind that not all DVI versions are the same.