Isn't this implant too close to my natural tooth? (photos)

yes, its very close to next tooth .  CBCT will give perfect idea. so its my advice to get CBCT done. 

From that angle its not possible to determine the proximity of the implant to the tooth, its the wrong type of xray to determine that.  My feeling is that it should not effect the long term stability or outcome of either the implant or the natural tooth.  I think your dentist is doing a good job from what I can see.

We are looking at a panoramic x-ray which has a 25% distortion rate associatedwith it.  It is possible to see something that actually does not exist.  In my opinion I would delay restoring the implants with crowns and monitor for the next few months.  If there are changes in the appearance of the xray theni would have it removed.  It may give you a chance to keep the tooth next to it.You could also be pro active and have a CT scan done.  This could give you a clearer picture of whether the implant is in contact with the tooth.

It might be. The best way to find out is to get a 3D view. If found to be close, you would need to watch the tooth with annual X-rays. 

There are 2 things to consider here, yes the implant its near the adjacent tooth, but the main reason you need to be worry, is not the implant, its the tooth next to it, as you can see in the picture, it has a shadow at the tip of the tooth, this is an  infection, and this may cause the implant to fail, and also the 2 adjacent teeth have receding bone, so sooner or later you will loose those teeth, also. you need to start thinking to set more implants on that area, i know you wanted to preserve those teeth with root canals, but as you can see there is an infection on top of them.