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We present a new construction of non-classical unitals from a classicalunital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals.The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with thefollowing three properties: 1. points of $\Pi$ are those of $PG(2,q^2)$; 2.lines of $\Pi$ are certain lines and conics of $PG(2,q^2)$; 3. the points in$U$ form a non-classical B-M unital in $\Pi$.  Our construction also works for the B-T unital, provided that conics arereplaced by certain algebraic curves of higher degree.

An alternative construction of B-M and B-T unitals in Desarguesian planes