Coded the previous sketch, played with few parameters and it is starting to get there ... I think. Will make another pass over this tomorrow and it should be good to go.
For what it's worth I have also gave a try to the standard Tooltip Control. In fact, that's where I started.

A regular tooltip looks like this -

It adds another layer to the windows stack, breaks the visual flow of the layout and still manages to tuck itself under the mouse cursor.

There's also a balloon version of the same, which looks like this -

It is marginally better, but just look at those lovely rounded cornersi. Eat that Apple. Real artists ship.

tl;dr - the standard Tooltip Control is not very good, so I had to come up with a custom replacement.