Yes, it is possible to do either via hard additional development or by changing "Displayed value" in object advanced settings for the object that is represented by the lookup http://prntscr.com/q50jbe. Please note that it is not preferred option since this value will be changed everywhere in the system where lookups based on "Contact" object are used.
Please also note that account field cannot be selected there since account is another lookup, but the system allows using only "String" data type fields when specifying "Displayed value", but you can add your custom hidden field and create a process that will fill this field in with the value of an account that is chosen for the contact specified.
