A column can have one and only one name. If your rowset contained only one row, then you could look at the row's Type column first and then change the column name appropriately for the select. If it contains multiple rows, it's simply not possible.
IF 1 = (SELECT COUNT(*) FROM Person P WHERE <where-criteria>) THEN
IF 'Individual' = (SELECT P.Type FROM Person P WHERE <where-criteria>) THEN
SELECT P.Amount AS Salary
FROM Person P
WHERE <where-criteria>
ELSE
SELECT P.Amount AS Profit
FROM Person P
WHERE <where-criteria>
END IF
ELSE
SELECT P.Amount AS SalaryOrProfit
FROM Person P
WHERE <where-criteria>
END IF
I think you might need to re-examine your design.