The Dirac equation arose out of the need for a quantum mechanical wave equation that treats position and time on an equal basis, just as relativity expects them to be. Now the Schrodinger equation is second order in space and first order in time rendering it non-relativistic.

One possibility for converting the non-relativistic Schrödinger equation for a free particle to a relativistic equation is to replace the classical energy $H=\frac{p^2}{2m}$by the relativistic energy $H=\sqrt{\mathbf p^2c^2+m^2c^4}$ and replacing $\mathbf p \rightarrow -i \hbar\nabla.$

Dirac’s solution was essentially to turn the problem on its head. Rather than trying to find the square root of an existing quantity, he postulated that the quantity inside the square root is the perfect square of an expression that is linear in the momentum, so that both space and time are involved as first order. $$H = -i \vec \alpha.\mathbf p + \beta m$$

My question is could we have also chosen a scenario where we had both space and time derivatives of second order instead of linear equations? Apart from possibly making the life complex, would there be any physical considerations not allowing this formulation?