I'll go even further: I wouldn't be surprised if the advent and spread of Zcash helped Bitcoin's value, at least for the next several years. They offer different kinds of values in some ways. For example, Bitcoin is very widely understood and widely built-on-top-of, and it has a strong tradition of evolving the protocol extremely carefully and slowly if at all. In contrast Zcash is newer and more experimental, could have "growing pains" kinds of bugs and security failures in the coming years (like Bitcoin did in its early years), and as we've recently blogged, we intend to make Zcash more of an evolving protocol than Bitcoin currently is. (But less quickly evolving than Ethereum currently is. )
My point is that there are use cases and trade-offs that better served by Bitcoin, at least for the forseeable future, and others that are better served by Zcash.
And, the two can interoperate and cooperate (the development teams behind Zcash and Bitcoin frequently cooperate on technical issues), and the two can grow in slightly different directions as they mature, doubling-down on the particular values that appeal most strongly to their most important user bases.
Also each one can learn from the other, copying both the ideas and technology from each other and also learning from what users and markets the other explores.
So, I wouldn't be at all surprised if the advent and the success of Zcash causes Bitcoin to become more valuable to more people.
Not that I — or anyone else — knows what will happen.