(warning –also a lack of any psychology diploma ahead )
There appears to be evidence that there are certain (basic) personality types. There also appears to be evidence that some personality traits are inherent from birth. Most of our personality develops over time though, due to age, social and environmental influences.
A flexible mind (capable of multitasking among other things) certainly has an influence on the development of our personality. I think it’s too simply put to say that personality starts to develop when we learn to multitask. However the idea that multitasking enables us to observe ourselves and so influence our behavior makes sense. I have come across this concept, in one way or another, in several schools of psychological thought. (can’t remember exact references at the moment) And in a few instances have been able to “observe” this process in myself.
So when does our personality start to develop? Given that there may or may not be basic personality types and inherent personality traits from birth. I think our mind/personality starts to develop as soon as our brain reaches a certain level of growth. (in a neurological sense, enough synaptic connections, capability to process sensory input and to retain memories and such)