While it has been rumoured for a while now thanks to data miners, Niantic finally acknowledged an upcoming feature that will let you have more one-on-one time with Pokemon. Here's Niantic, explaining how it will generally work:
The next addition to Pokémon GO will be Buddy Pokémon. You will be able to pick your favourite Pokémon from your collection to become your buddy, opening up unique in-game rewards and experiences. Buddy Pokémon will appear alongside your Trainer avatar on your profile screen, adding helpful bonuses such as awarding Candy for walking together. You'll also have the ability to change your Buddy Pokémon at any time.
There's no specific ETA detailed by Niantic, but they say it's "just around the corner".