What unique challenges/limitations will I face if I start a career as a pilot at 45 years old?
Time

All other things aside all commercial part 121 pilots have a finite end date, starting at 45 means you are going to be ~20 years behind the young guys. Since the FAA mandates retirement from part 121 operations at 65 you have 20 years left but let's look at that a bit more practically. Assuming you pay for all your own training and don't need to go the CFI route to build hours you could conceivably finish your 1500 ATP hours in a year's time if you flew full time and the weather gods were nice to you. However that's not really realistic for all kinds of reasons, weather, mechanical, practical etc. So let's say you get there in a modest 2 years. Now you're 47... 

At 47 you are applying to the airlines, still need to actually pass your ATP exams, and then they need to type you into one of their aircrafts, that's simulator time and ground school time for the airline which is by no means free and takes time - this means you are getting older. They are investing in you and in their eyes they can now only get 17 years or less out of that investment unlike the young guys they may have for longer. A regional may be interested but expect long grueling days and pay on the lower side of things. 

You are also starting at the bottom of the seniority tree. If you are accustomed to a nice self sufficient lifestyle now you, should not expect that working at a regional for the first few years. 



There are lots of other flying jobs that are not bound by the 65 year old limit, they may be a bit more transient in nature but you can still get into a high performance airplane and be having fun. You can fly for a part 135 charter operation, be a flight instructor, tow banners, tow gliders, crop dust, fly sight seeing flights, etc. One job that I have spoken to a few professional pilots about that seems like a sweet deal is being on staff flying a corporate jet for a company, the hours tend to be decent, you end up flying some fast metal and going cool places.