What kind of base are you thinking about? Is it to be incorporated into the design or is it something that can be done separate?
I know you can mold a three-dimensional piece and include a base easily enough. You can also blow such a piece as you’ve described where the glass cools enough to maintain its shape, but not enough to melt. You can also carve (sandblast) to get a base. You can glue the base to the form. You can make a non-glass base (metal, wood, etc.) and adhere the two.
A friend makes the “vertical piece”, pours a mold around the glass to maintain the shape, then carves out the base in the mold and refires with additional glass for the base.