Having built several large apps which are live in the Store in both languages, I'm sticking with Obj-C for 2016. Give Swift another year before going all in.
I think one of the biggest misconceptions with the migration to Swift is that it will be a lot easier to be an iOS developer with no experience. After teaching new to programming students iOS development with both Objective-C and Swift, the actual syntax mattered very little - the underlying iOS APIs were difficult to understand.
You'll still find that many API tutorials are written with Objective-C, so even if you don't code in that language, you'll need to translate in your head as you read how to hook up X or Y to Z. In addition, a lot of stack overflow questions for common iOS programming questions are in Objective-C, so you'll at least need to know how to read it.
Even if you wanted to learn Objective-C all the good versions of learning materials are at iOS 7 and below. Not a good choice since it is very likely none of that stuff works cleanly anymore. So, go with Swift. Yes, you will need to be familiar with Objective-C on an as needed basis, but if you learn Swift and iOS programming it will be a translation and not that bad.
The reality is if you can't be proficient with Objective-c no one will hire you at this point. The more time passes, the more Objective-c code will become legacy, and the less important this will be.
That being said definitely start with Swift. It is an excellent language and is the future of the iOS platform.
At the very least, your need to be able to reason about both languages. Refusing to learn one of them is a disservice to yourself as an iOS dev IMO. Denying that would be silly. If you haven't taken a look at Swift, there is no better time than now to start.