It depends upon the ghost. And, it depends on how you define “alive.”
I believe that the spirit of the person remains alive. (The body of that person, in this plane of existence, is lifeless. That’s a semantic issue.)
In general, I believe that ghosts are as alive as you and I are. Despite that, we usually refer to ghosts as spirits of the dead.
One ghostly exception: some hauntings don’t seem to be actual ghosts. They’re more like energy imprints – often called “residual energy hauntings” – that linger at a location after something dramatic happened.
It’s like when you walk into a room where two people recently argued. Sometimes, you can still feel the crackle of hostile energy in the room.
For more about this topic, see my articles, Hauntings – residual or ‘real’? and Residual energy imprint hauntings.
Some people (and some ghosts) object to the “living v. dead” distinction.
When that’s an issue, I stick to words like: ghost, spirit, deceased, and entity, or I always refer to the ghost by his or her assumed name.
For more, ghost-related terms and phrases, see my article, What are some words that refer to ghosts?
For example, although we’re not sure the most famous ghost at the Myrtles Plantation is actually called “Chloe,” we use that name for her anyway.
When I’m ghost hunting at the Myrtles, and I want to speak with that ghost, I’ll say, “Chloe, I’d like to talk with you,” and then say or ask whatever is on my mind.
Also, most ghosts seem to find very little humor related to the subject of death. The jokes and puns at Disney’s Haunted Mansion attraction may seem funny to you. (They strike me funny, too.)
However, avoid morbid jokes during ghost investigations. (I’m talking about things like “dying to meet you” and “feeling dead tired,” and so on.)
Offending ghosts is a very bad idea, if you’re interested in learning more about them, or want their cooperation.
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Learn more about ghosts and haunted places at Fiona’s YouTube channel, Ghost Hunting with Fiona Broome.