Ken,
Thanks for posting this - it is a really interesting bottle. If I did not know where it came from, then I might first be wondering whether it is modern or not because it is simply so atypical of a 19th century bottle. It would be helpful to know more about the other bottles and shards that you found there - specifically their age.
I think most collectors would say:
- the color suggests a hair bottle.
- age wise, late 19th century but could even be earlier 20th century
- the fancy shape suggests possibly a bitters bottle.
- the simple sheared and refired lip is entirely unusual and makes me wonder whether it could be European.
No matter what the consensus of opinion, you are not going any further with your investigation unless you are lucky enough to find an example with labels.
Years ago, I owned a bitters bottle of a similar shape that had full labels. I believe it was German. Once I recall the name, I will report back with further info.