There are different kinds of immortality.

They could be immortal like Deadpool. I don't think that's the most interesting option in a cosmic horror setting.

But why would the elder god grant the gift of healing? Why would he prevent the ravages of time? A thousand years from now, the immortal investigator will be no more than a surviving consciousness in a dried-up eye, enshrined in a relic, gazing out at the world for all eternity.

Or what if immortality was just the knowledge of a way to combat the ravages of time, but not violent death? What would be the price to pay? Absorbing the vital energy of other humans (or rarer, more dangerous entities) at regular intervals? Transfering their consciousness from other bodies ?

What would the investigators do to protect themselves physically and organize their survival? They'd need the means, the wealth, and reliable people at their service. How could they ensure their loyalty? How be sure they won't betray them to acquire their position or their secrets of eternel life?

At what point will they attract the attention of other dark cults or people acting to save humanity, like investigators or government agencies like Delta Green?

At some point, you could even start a new campaign in which their immortal characters are the hidden leaders of a dark cult that threatens the survival of mankind