Not a strange thought, at all, with several interesting solutions described over the years. The information below is not presented as a construction idea: elements are #14 wire and lengths are approximate. The antenna is simulated over "real" ground of "medium" conductivity (Sommerfeld model), with the lower set of elevated radials .05$\lambda$ above ground and .01$\lambda$ vertical separation between the radial sets to prevent shorts. More engineering would have to be done to result in a practical design.
Just as you would with a horizontal yagi, a lengthened vertical parasitic reflector (wire 6) could be added to a driven vertical (wire 1):
The horizontal spacing between the vertical elements is 0.15$\lambda$, the driven vertical and the radials (wires 2-4 and 7-10) are 0.24$\lambda$ long and the parasitic element (wire 6) is 9% longer than the driven element, producing some gain and useful directivity (elevation plane shown):
Alternatively, you could change the electrical length of the parasitic element by adding reactance to it. Instead of changing the physical length of the parasitic vertical, a 550nH inductor, similar to what one might use as a loading coil for a matching network, is added to the base (note the square at the base of element 6):
again with some gain and useful directivity (elevation plane shown here):
Sadly, modeling does not show useful directivity when replacing the inductive reactance with capacitance in an effort to make the loaded parasitic reflector into a director, so there is some limit to the flexibility of this approach. However, one could swap the driven and loaded connections to reverse the pattern.