Variant of ridge regression loss function


We can create variants of the loss function, especially of ridge regression by adding more regularizer terms. One of the variants I saw in a book is given below

$ min_{w \in \mathbf{R}^d} \ \ \alpha.||w||^2 + (1-\alpha).||w||^4 + C||y-X^T.w||^2$

where $ y \in \mathbf{R^n}, s \in \mathbf{R^d}, X \in \mathbf{R^{dxn}}$ and $ C$ a regularization parameter $ \in \mathbf{R}$ and $ \alpha \in [0,1]$

My question is how does change in $ \alpha$ affects our optimization problem? and how does generally adding more regularizers help? why is not one regularizer enough?