How to uninstall this fancy bathroom faucet?
You should be able to remove that small cap at the very top of the cold water valve. Try to pry it off with a small screwdriver. There should be a brass screw inside there, remove it to remove the handle. Then the cone should lift off or unscrew counterclockwise. That should expose the valve stem. There's packing and washers that would need to be replaced or you might have to replace the valve stem. Look for a good plumbing supple store. Remember to turn off the water to that faucet before removing any parts. Water is probably seeping out through the top and dripping out the bottom.