Shattered

Shattered is a main story quest in Fallout 4, following Dangerous Minds.

Lounging around at the Memory Den
After killing Conrad Kellogg, loot the cybernetic brain augmenter to delve deep into his memories. Gather it from Fort Hagen (as part of the previous quest, Reunions) if it hasn't been taken already.

Follow Nick Valentine's advice and locate Doctor Amari. This involves a trip into Boston, through the ruins of the Financial District, to the rough-and-tumble fortified settlement of Goodneighbor. Greet the locals, then head to the Memory Den, owned by Irma.

Down in the basement, converse with Doctor Amari. After Nick Valentine volunteers to help out, the Sole Survivor is invited to sit in the lounger to experience snippets of Kellogg's past.

Memories
The Sole Survivor wanders through the ethereal strands of Kellogg's memory. Stepping from memory to memory, scenes play out revealing pieces of his life. In most of the memories, various things can be activated to trigger additional monologues, where Kellogg will go into more detail about them.

The memories reveal that Kellogg does indeed work for the Institute, took Shaun from Vault 111, and that they use some kind of teleportation to operate freely. The final scene reveals that Kellogg's current mission is to hunt down a runaway scientist named Virgil, who likely escaped into the Glowing Sea.

Leave Memory
Upon conclusion of the final scene, Doctor Amari will instruct the Sole Survivor to interact with the television to leave the memory. Back in the Memory Den, speak to Amari about Virgil.

When leaving the Memory Den, there is an optional conversation with Nick Valentine where he will first seem to have been taken over by Kellogg's memory, only for Nick to seemingly regain control. He will also offer additional advice about hunting Virgil.

Companion reactions
Every companion except Curie likes when the memory playback is completed. Curie is indifferent, and Nick is always present for the event to trigger.