The ending sounds unfinished because you haven't harmonized it. This is my opinion of course, but if your piece isn't specifically a "solo", or if you are not executing a cadenza-like passage (something Beethoven is known for, especially in his Piano Concertos), I would probably just avoid long drawn out solos. Solos have the advantage of being a tension breaker, tension builder, or "the main character" of a piece. But this short piece needs to take advantage of the time it has by communicating as much as it can in such a short amount of time. The solos waste valuable time.
And you don't need complex harmony. When I say "simple functional" harmony, that isn't a criticism. Chopin's Fantaisie Impromptu and Beethoven's 5th Symphony use simple functional harmony and they are masterpieces. In fact, you wouldn't want overly complex harmony since your piece is highly melodic, and complex harmony would take away from the melody.