An interesting question about a man who was "a Scot by birth, a doctor
of physic by profession, a fiery Puritan in faction".
First, a brief biography of Leighton. Born in 1587 in Scotland,
Leighton was first a physician before becomeing a full time Puritan
preacher and pamphleteer. He rose to infamy after publishing 'Zion's
Plea Against Prelacy: An Appeal to Parliament' in 1630, during the
decidedly Anglo-Catholic reign of Charles I, This was an attack on the
the bishops which caused outrage in the church. He was sentenced by
Archbishop Laud's High Commission Court to public whipping, to having
the letters 'SS' branded on him (for 'Sower of Sedition'), and having
his ears cut off and his nostrils slit. Leighton finally died in 1644.
With rather delicious irony, Leighton's son Robert went on to becomne
an Archbishop himself.
As for his writings, and his connection with Shakespeare - there is a
frustrating lack of concrete evidence here. Unsurprisingly, given what
happened to Leighton, the majority of seditious pamphlets were
published anonymously. If Leighton did publish other works, we won't
know for sure that they were written by him.
However, one pamphlet which has been attributed to Leighton may have a
Shakespearean connection. 'A Short Treatise against Stage-Playes',
published in 1625, "a brief and exceedingly businesslike enumeration
of the chief arguments against the drama", is attributed to our man in
the British Library catalogue, and is given as the author in the 1973
Garland Publishing edition (see below). Unfortunately, I haven't been
able to find a synopsis of the book online other than the fact that
"in [its] twenty-eight pages may be found the whole gist of [fellow
puritan William Prynne's] Histriomastix." The latter book contains a
single reference to Shakespeare, that "Shackpeers Plaies are printed
in the best Crowne paper, far better than most Bibles."
All references in the above paragraph from
The Puritan Attack upon the Stage
http://www.bartleby.com/216/1416.html#note95 for reference to our man)
Having said all that, of course, it seems unlikely that Leighton, a
London-based Puritan activist, would not have been aware of
Shakespeare's plays and active in the controversy surrounding the
morality of the stage which was such a hot topic in Puritan circles in
Stuart England, leading finally to the closing of all theatres in
I hope this is of use to you and that the links below can spread
further light on Leighton and his times.
The High Commission Court
National Portrait Gallery: Alexander Leighton Portrait
The Puritan Paradox
If you need any more information, please ask for clarification.