Yes, the HasExited will always be true in your case.
According to MSDN,
"A value of true for HasExited indicates that the associated process has terminated, either normally or abnormally.[...]A process can terminate independently of your code. If you started the process using this component, the system updates the value of HasExited automatically, even if the associated process exits independently."
However, if your process crashes and terminates before your timeout, then your code will set it as completed anyway. Maybe you should check the exit code, but it can have different meanings for each process:
if (process.ExitCode != 0)
{
this.completed = false;
}
For crashes, there are some approaches here and here, but generally you can't detect crashes for all processes.