This is because iex master process is trapping exits, which means it doesn’t just crash when a linked process crashes. Instead, it receives a message, that is handled in form of error being printed to the user.

Your code will work as you’d expect if you didn’t start it from iex. Even in iex you can spawn an intermediary process to observe the behavior as you’d expect:

iex(2)> defmodule Greeter do                                                                                                                                              
...(2)>   def run do
...(2)>     task = Task.async(&greeter/0)
...(2)>     Task.await(task, 10_000)
...(2)>     IO.puts("work is done")
...(2)>   end
...(2)>
...(2)>   defp greeter do
...(2)>     Process.sleep(1000)
...(2)>     IO.puts("Hello")
...(2)>     greeter()
...(2)>   end
...(2)> end
warning: redefining module Greeter (current version defined in memory)
iex:2

{:module, Greeter,
<<70, 79, 82, 49, 0, 0, 6, 208, 66, 69, 65, 77, 65, 116, 85, 56, 0, 0, 0, 221,                                                                                           
  0, 0, 0, 23, 14, 69, 108, 105, 120, 105, 114, 46, 71, 114, 101, 101, 116,                                                                                              
  101, 114, 8, 95, 95, 105, 110, 102, 111, 95, ...>>, {:greeter, 0}}                                                                                                     
iex(3)> spawn fn -> Greeter.run() end
#PID<0.145.0>
Hello
Hello
Hello
Hello
Hello
Hello
Hello
Hello
Hello
iex(4)>