```python
from collections import deque
from typing import Dict, Optional

class JobQueue:
    def __init__(self):
        self._queue: deque = deque()
        self._jobs: Dict[str, dict] = {}
        self._results: Dict[str, dict] = {}

    def add_job(self, job_id: str, data: dict) -> str:
        if job_id in self._jobs:
            raise ValueError(f"Job ID '{job_id}' already exists")
        self._jobs[job_id] = data
        self._queue.append(job_id)
        return job_id

    def get_result(self, job_id: str) -> Optional[dict]:
        return self._results.get(job_id)

    def process(self):
        while self._queue:
            job_id = self._queue.popleft()
            if job_id in self._jobs:
                job_data = self._jobs.pop(job_id)
                # Simulate job completion
                self._results[job_id] = {"status": "completed", "data": job_data}
```