Stir blueberries, sugar (or other sweetener), lemon juice, and water in a medium saucepan. Place over medium heat and cook, stirring occasionally, until the mixture boils and the blueberries start to burst 6 to 9 minutes.
Remove from the heat and let cool. (Transfer to a metal bowl and set in the fridge, or submerge the saucepan in an ice bath to speed up this process.)
Puree the blueberry mixture until completely smooth using a blender, food processor, or immersion blender.
Set a fine mesh sieve over a medium bowl. Pour the puree into the sieve and run through with a silicone spatula.
Transfer to a jar and chill. Serve cold or at room temperature.
