The price will only increase if there is demand. I think the existence of a single coin does not affect the price much. According to the simple rule of economics, the more buyers of a good, the more its price increases. Therefore, whether Bitcoin stands alone or not, it can proceed in a similar way.
There are many cryptocurrencies, but which one can you trust as much as Bitcoin? Your answer to this question will be similar to the answer you would give to the question asked in the topic.
in every market, competition is important to know the difference of their role. however, if there is no competition, i guess, people won't appreciate much about its existence.
and how people look at the demand factor may be different.
if bitcoin was the only cryptocurrency you can invest in, its scarcity might be highlighted more
a lot of investors resort to altcoins because bitcoin is too expensive now, basically altcoins gives people choices but without other choices bitcoin might stand as unique and rare
everyone would flock into bitcoin which probably clog the system making confirmation times longer