There exist two approaches to the conformal limit mechanism: first was defined by Gaiotto using Analysis techniques and the method of computing was first established for the Hitchin Section and Opers in 2016. The second approach to conformal limits as algebraic shifts via extension classes of vector bundles was established by Dumitrescu and Mulase in 2017 for the lagrangians Hitchin section and opers. In this talk I will report on work in progress with Jennifer Brown and Motohico Mulase of the algebraic approach of conformal limits to a family of Lagrangians covering the Dolbeault and the De Rham moduli space of Higgs bundles and irreducible connections over a curve in rank 2.