Disclaimer: I don't own Doctor Who.
You spend each day closely watching every person you see, fully prepared to run.
It's not the type of running you'd do for recreation or exercise or because you're late. It's the type of running that comes with someone, or some thing, chasing you. The fear, the exhilarating rush of adrenaline and speed that comes with running from someone, or some thing, dangerous.
Most people would find this exceedingly odd, wondering why in the world you would want to be chased by danger, but you know better. You know that if that day comes, he will be there. He will grab your hand and say "Run!" and you will realize that you were born to do this.
You will realize that you were born to run, born to feel that energy blossom in your veins. It's like a drug, so addicting that you can't possibly get enough of it. He grins at you as you turn a corner together. You fail to hide your own smile from him.
The people standing around stare, but they brush the both of you off easily. They don't know what's happening or the magnitude of what you just gained. You escape the danger and you know what is coming. He invites you into that magical blue box and you are completely content to travel in the sky, leaving behind nothing but stardust.
You see aliens and monsters on other worlds, none of them paying any mind to your odd customs or clothes. The man from the box takes you to dozens of places among the stars and you help him save planets and people. You see snow on a moon, plants in the sky, and sunlight underground. You see the unfathomable courage that is the people of the universe, living their magnificent day to day lives. Sometimes they are happy. Sometimes they are sad. You always make a point to give the crying child a cookie or the homeless beggar your extra money. The man thanks you profusely when you help him pick up the books that he has dropped. The woman in the shop smiles gratefully when you overpay her for the souvenir and insist that she keep it.
You know now that you have been fundamentally changed. From the smallest atom in your body, you are different. You have seen the expanse of the universe, the immense beauty and the tragic loss. You have seen the people of the universe live their lives, lives that aren't so different from yours. You can see now that everyone is the same, no one is different, and everyone is important. You find yourself like the Doctor, pleased in so many ways at this development. You are excited at this change; you know it is for the better.
You find yourself completely believing in this man, this Time Lord. You believe that he can save everyone. You believe that he will never let any harm come to you. You believe that planets will be safe in the sky all because of him. And your beliefs ring true.
Sometimes, not all the time, someone dies. You mourn their death along with the Doctor, taking no consolation in the fact that there was nothing that could have been done. The both of you keep going while you silently accept the fact that it may be you one day.
When that day comes, you are prepared. Perhaps you die. Perhaps you must leave for a reason that is beyond your control. Whatever the circumstances, you know that it isn't truly the end for you and certainly not for him. If you die, he will remember you. If you live, he will never forget. You are changed in the most spectacular ways and you spread this change across Earth, hoping that just a little bit of the Doctor will change the whole of the human race.
If nothing else, you owe him that much.