Boughtcar second hand, has low mileage for year 1998, it had been standing for a few months without being started and l had a problem with battery holding it is charge so have had the alternator replaced andthe car running great. Problem is the left indicator light stays on even with no key in the ignition, right indicator working as normal.

Inside the turn signal ''stalk'' there r switches for many things. High beams, low beams, turn signals, horn, lights on & off, maybe more.
When ur car is turned off, & the key is OUT of the ignition, ur headlights can still be ON. This means that electricity is flowing through the ''stalk''.

My guess. & l have no proof or experience on the 206 stalk.. is that u have a small, but nagging, short circuit inside the stalk. This stalk is IMPOSSIBLE TO REPAIR, & here l do have experience with the 205, & 405. Many springs, levers, copper plates, plastic separators..after u DRILL the rivets out of the stalk, it all falls apart in 10 pieces on the table.

Replace the stalk. Get one from a breaker yard, & learn how to remove it from an old car. Or, buy a new one. It will be easy to replace with the correct screw driver (maybe a torx) in ur driveway.

Hope this solves the problem for you. (see if a few sharp thumps with ur fist will stop the little short circuit??)

PS: ur previous battery problem. When a battery is not kept charged up, regularly, it loses its ability to hold a charge at all. Your problem may not have been the alternator. You may still have a weak battery. Time will tell. or maybe u ALSO needed & alternator due to its age?